(************************************************************************)
(*         *   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 Rtrigo1.
Require Import Ranalysis1.
Require Import PSeries_reg.
Require Import Div2.
Require Import Even.
Require Import Max.
Require Import Omega.
Local Open Scope nat_scope.
Local Open Scope R_scope.

Definition E1 (x:R) (N:nat) : R :=
  sum_f_R0 (fun k:nat => / INR (fact k) * x ^ k) N.

Lemma E1_cvg : forall x:R, Un_cv (E1 x) (exp x).
forall x : R, Un_cv (E1 x) (exp x)
Proof.
forall x : R, Un_cv (E1 x) (exp x)
  intro; unfold exp; unfold projT1.
x:R
Un_cv (E1 x) (proj1_sig (exist_exp x))
  case (exist_exp x); intro.
x, x0:R
forall e : exp_in x x0,
Un_cv (E1 x)
  (proj1_sig (exist (fun l : R => exp_in x l) x0 e))
  unfold exp_in, Un_cv; unfold infinite_sum, E1; trivial.
Qed.

Definition Reste_E (x y:R) (N:nat) : R :=
  sum_f_R0
  (fun k:nat =>
    sum_f_R0
    (fun l:nat =>
      / INR (fact (S (l + k))) * x ^ S (l + k) *
      (/ INR (fact (N - l)) * y ^ (N - l))) (
        pred (N - k))) (pred N).

Lemma exp_form :
  forall (x y:R) (n:nat),
    (0 < n)%nat -> E1 x n * E1 y n - Reste_E x y n = E1 (x + y) n.
forall (x y : R) (n : nat),
(0 < n)%nat ->
E1 x n * E1 y n - Reste_E x y n = E1 (x + y) n
Proof.
forall (x y : R) (n : nat),
(0 < n)%nat ->
E1 x n * E1 y n - Reste_E x y n = E1 (x + y) n
  intros; unfold E1.
x, y:R
n:nat
H:(0 < n)%nat
sum_f_R0 (fun k : nat => / INR (fact k) * x ^ k) n *
sum_f_R0 (fun k : nat => / INR (fact k) * y ^ k) n -
Reste_E x y n =
sum_f_R0 (fun k : nat => / INR (fact k) * (x + y) ^ k)
  n
  rewrite cauchy_finite.
x, y:R
n:nat
H:(0 < n)%nat
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun p : nat =>
      / INR (fact p) * x ^ p *
      (/ INR (fact (k - p)) * y ^ (k - p))) k) n +
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      / INR (fact (S (l + k))) * x ^ S (l + k) *
      (/ INR (fact (n - l)) * y ^ (n - l)))
     (Init.Nat.pred (n - k))) (Init.Nat.pred n) -
Reste_E x y n =
sum_f_R0 (fun k : nat => / INR (fact k) * (x + y) ^ k)
  n
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  unfold Reste_E; unfold Rminus; rewrite Rplus_assoc;
    rewrite Rplus_opp_r; rewrite Rplus_0_r; apply sum_eq;
      intros.
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
sum_f_R0
  (fun p : nat =>
   / INR (fact p) * x ^ p *
   (/ INR (fact (i - p)) * y ^ (i - p))) i =
/ INR (fact i) * (x + y) ^ i
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  rewrite binomial.
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
sum_f_R0
  (fun p : nat =>
   / INR (fact p) * x ^ p *
   (/ INR (fact (i - p)) * y ^ (i - p))) i =
/ INR (fact i) *
sum_f_R0
  (fun i0 : nat => C i i0 * x ^ i0 * y ^ (i - i0)) i
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  rewrite scal_sum; apply sum_eq; intros.
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
/ INR (fact i0) * x ^ i0 *
(/ INR (fact (i - i0)) * y ^ (i - i0)) =
C i i0 * x ^ i0 * y ^ (i - i0) * / INR (fact i)
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  unfold C; unfold Rdiv; repeat rewrite Rmult_assoc;
    rewrite (Rmult_comm (INR (fact i))); repeat rewrite Rmult_assoc;
      rewrite <- Rinv_l_sym.
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
/ INR (fact i0) *
(x ^ i0 * (/ INR (fact (i - i0)) * y ^ (i - i0))) =
/ (INR (fact i0) * INR (fact (i - i0))) *
(x ^ i0 * (y ^ (i - i0) * 1))
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact i) <> 0
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  rewrite Rmult_1_r; rewrite Rinv_mult_distr.
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
/ INR (fact i0) *
(x ^ i0 * (/ INR (fact (i - i0)) * y ^ (i - i0))) =
/ INR (fact i0) * / INR (fact (i - i0)) *
(x ^ i0 * y ^ (i - i0))
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact i0) <> 0
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact (i - i0)) <> 0
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact i) <> 0
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  ring.
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact i0) <> 0
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact (i - i0)) <> 0
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact i) <> 0
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  apply INR_fact_neq_0.
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact (i - i0)) <> 0
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact i) <> 0
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  apply INR_fact_neq_0.
x, y:R
n:nat
H:(0 < n)%nat
i:nat
H0:(i <= n)%nat
i0:nat
H1:(i0 <= i)%nat
INR (fact i) <> 0
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  apply INR_fact_neq_0.
x, y:R
n:nat
H:(0 < n)%nat
(0 < n)%nat
  apply H.
Qed.

Definition maj_Reste_E (x y:R) (N:nat) : R :=
  4 *
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * N) /
    Rsqr (INR (fact (div2 (pred N))))).

(**********)
Lemma div2_double : forall N:nat, div2 (2 * N) = N.
forall N : nat, Nat.div2 (2 * N) = N
Proof.
forall N : nat, Nat.div2 (2 * N) = N
  intro; induction  N as [| N HrecN].
Nat.div2 (2 * 0) = 0%nat
N:nat
HrecN:Nat.div2 (2 * N) = N
Nat.div2 (2 * S N) = S N
  reflexivity.
N:nat
HrecN:Nat.div2 (2 * N) = N
Nat.div2 (2 * S N) = S N
  replace (2 * S N)%nat with (S (S (2 * N))).
N:nat
HrecN:Nat.div2 (2 * N) = N
Nat.div2 (S (S (2 * N))) = S N
N:nat
HrecN:Nat.div2 (2 * N) = N
S (S (2 * N)) = (2 * S N)%nat
  simpl; simpl in HrecN; rewrite HrecN; reflexivity.
N:nat
HrecN:Nat.div2 (2 * N) = N
S (S (2 * N)) = (2 * S N)%nat
  ring.
Qed.

Lemma div2_S_double : forall N:nat, div2 (S (2 * N)) = N.
forall N : nat, Nat.div2 (S (2 * N)) = N
Proof.
forall N : nat, Nat.div2 (S (2 * N)) = N
  intro; induction  N as [| N HrecN].
Nat.div2 (S (2 * 0)) = 0%nat
N:nat
HrecN:Nat.div2 (S (2 * N)) = N
Nat.div2 (S (2 * S N)) = S N
  reflexivity.
N:nat
HrecN:Nat.div2 (S (2 * N)) = N
Nat.div2 (S (2 * S N)) = S N
  replace (2 * S N)%nat with (S (S (2 * N))).
N:nat
HrecN:Nat.div2 (S (2 * N)) = N
Nat.div2 (S (S (S (2 * N)))) = S N
N:nat
HrecN:Nat.div2 (S (2 * N)) = N
S (S (2 * N)) = (2 * S N)%nat
  simpl; simpl in HrecN; rewrite HrecN; reflexivity.
N:nat
HrecN:Nat.div2 (S (2 * N)) = N
S (S (2 * N)) = (2 * S N)%nat
  ring.
Qed.

Lemma div2_not_R0 : forall N:nat, (1 < N)%nat -> (0 < div2 N)%nat.
forall N : nat, (1 < N)%nat -> (0 < Nat.div2 N)%nat
Proof.
forall N : nat, (1 < N)%nat -> (0 < Nat.div2 N)%nat
  intros; induction N as [| N HrecN].
H:(1 < 0)%nat
(0 < Nat.div2 0)%nat
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
(0 < Nat.div2 (S N))%nat
  -
H:(1 < 0)%nat
(0 < Nat.div2 0)%nat elim (lt_n_O _ H).
  -
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
(0 < Nat.div2 (S N))%nat cut ((1 < N)%nat \/ N = 1%nat).
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
(1 < N)%nat \/ N = 1%nat -> (0 < Nat.div2 (S N))%nat
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
(1 < N)%nat \/ N = 1%nat
    {
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
(1 < N)%nat \/ N = 1%nat -> (0 < Nat.div2 (S N))%nat intro; elim H0; intro.
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H0:(1 < N)%nat \/ N = 1%nat
H1:(1 < N)%nat
(0 < Nat.div2 (S N))%nat
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H0:(1 < N)%nat \/ N = 1%nat
H1:N = 1%nat
(0 < Nat.div2 (S N))%nat
      +
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H0:(1 < N)%nat \/ N = 1%nat
H1:(1 < N)%nat
(0 < Nat.div2 (S N))%nat destruct (even_odd_dec N) as [Heq|Heq].
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H0:(1 < N)%nat \/ N = 1%nat
H1:(1 < N)%nat
Heq:even N
(0 < Nat.div2 (S N))%nat
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H0:(1 < N)%nat \/ N = 1%nat
H1:(1 < N)%nat
Heq:odd N
(0 < Nat.div2 (S N))%nat
        *
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H0:(1 < N)%nat \/ N = 1%nat
H1:(1 < N)%nat
Heq:even N
(0 < Nat.div2 (S N))%nat rewrite <- (even_div2 _ Heq); apply HrecN; assumption.
        *
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H0:(1 < N)%nat \/ N = 1%nat
H1:(1 < N)%nat
Heq:odd N
(0 < Nat.div2 (S N))%nat rewrite <- (odd_div2 _ Heq); apply lt_O_Sn.
      +
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H0:(1 < N)%nat \/ N = 1%nat
H1:N = 1%nat
(0 < Nat.div2 (S N))%nat rewrite H1; simpl; apply lt_O_Sn. }
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
(1 < N)%nat \/ N = 1%nat
    inversion H.
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
H1:1%nat = N
(1 < 1)%nat \/ 1%nat = 1%nat
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
m:nat
H1:(2 <= N)%nat
H0:m = N
(1 < N)%nat \/ N = 1%nat
    right; reflexivity.
N:nat
H:(1 < S N)%nat
HrecN:(1 < N)%nat -> (0 < Nat.div2 N)%nat
m:nat
H1:(2 <= N)%nat
H0:m = N
(1 < N)%nat \/ N = 1%nat
    left; apply lt_le_trans with 2%nat; [ apply lt_n_Sn | apply H1 ].
Qed.

Lemma Reste_E_maj :
  forall (x y:R) (N:nat),
    (0 < N)%nat -> Rabs (Reste_E x y N) <= maj_Reste_E x y N.
forall (x y : R) (N : nat),
(0 < N)%nat ->
Rabs (Reste_E x y N) <= maj_Reste_E x y N
Proof.
forall (x y : R) (N : nat),
(0 < N)%nat ->
Rabs (Reste_E x y N) <= maj_Reste_E x y N
  intros; set (M := Rmax 1 (Rmax (Rabs x) (Rabs y))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs (Reste_E x y N) <= maj_Reste_E x y N
  apply Rle_trans with
    (M ^ (2 * N) *
      sum_f_R0
      (fun k:nat =>
        sum_f_R0 (fun l:nat => / Rsqr (INR (fact (div2 (S N)))))
        (pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs (Reste_E x y N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold Reste_E.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs
  (sum_f_R0
     (fun k : nat =>
      sum_f_R0
        (fun l : nat =>
         / INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l)))
        (Init.Nat.pred (N - k))) (Init.Nat.pred N)) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
        Rabs
        (sum_f_R0
          (fun l:nat =>
            / INR (fact (S (l + k))) * x ^ S (l + k) *
            (/ INR (fact (N - l)) * y ^ (N - l))) (
              pred (N - k)))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs
  (sum_f_R0
     (fun k : nat =>
      sum_f_R0
        (fun l : nat =>
         / INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l)))
        (Init.Nat.pred (N - k))) (Init.Nat.pred N)) <=
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         / INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l)))
        (Init.Nat.pred (N - k)))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         / INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l)))
        (Init.Nat.pred (N - k)))) (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply
    (Rsum_abs
      (fun k:nat =>
        sum_f_R0
        (fun l:nat =>
          / INR (fact (S (l + k))) * x ^ S (l + k) *
          (/ INR (fact (N - l)) * y ^ (N - l))) (
            pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         / INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l)))
        (Init.Nat.pred (N - k)))) (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
        sum_f_R0
        (fun l:nat =>
          Rabs
          (/ INR (fact (S (l + k))) * x ^ S (l + k) *
            (/ INR (fact (N - l)) * y ^ (N - l)))) (
              pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         / INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l)))
        (Init.Nat.pred (N - k)))) (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        (/ INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l))))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        (/ INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l))))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
Rabs
  (sum_f_R0
     (fun l : nat =>
      / INR (fact (S (l + n))) * x ^ S (l + n) *
      (/ INR (fact (N - l)) * y ^ (N - l)))
     (Init.Nat.pred (N - n))) <=
sum_f_R0
  (fun l : nat =>
   Rabs
     (/ INR (fact (S (l + n))) * x ^ S (l + n) *
      (/ INR (fact (N - l)) * y ^ (N - l))))
  (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        (/ INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l))))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply
    (Rsum_abs
      (fun l:nat =>
        / INR (fact (S (l + n))) * x ^ S (l + n) *
        (/ INR (fact (N - l)) * y ^ (N - l)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        (/ INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l))))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
        sum_f_R0
        (fun l:nat =>
          M ^ (2 * N) * / INR (fact (S l)) * / INR (fact (N - l)))
        (pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        (/ INR (fact (S (l + k))) * x ^ S (l + k) *
         (/ INR (fact (N - l)) * y ^ (N - l))))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   Rabs
     (/ INR (fact (S (l + n))) * x ^ S (l + n) *
      (/ INR (fact (N - l)) * y ^ (N - l))))
  (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun l : nat =>
   M ^ (2 * N) * / INR (fact (S l)) *
   / INR (fact (N - l))) (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 (S (n0 + n))) * x ^ S (n0 + n) *
   (/ INR (fact (N - n0)) * y ^ (N - n0))) <=
M ^ (2 * N) * / INR (fact (S n0)) *
/ INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  repeat rewrite Rabs_mult.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 (S (n0 + n)))) *
Rabs (x ^ S (n0 + n)) *
(Rabs (/ INR (fact (N - n0))) * Rabs (y ^ (N - n0))) <=
M ^ (2 * N) * / INR (fact (S n0)) *
/ INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  do 2 rewrite <- RPow_abs.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 (S (n0 + n)))) * Rabs x ^ S (n0 + n) *
(Rabs (/ INR (fact (N - n0))) * Rabs y ^ (N - n0)) <=
M ^ (2 * N) * / INR (fact (S n0)) *
/ INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rabs_right (/ INR (fact (S (n0 + n))))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) * Rabs x ^ S (n0 + n) *
(Rabs (/ INR (fact (N - n0))) * Rabs y ^ (N - n0)) <=
M ^ (2 * N) * / INR (fact (S n0)) *
/ INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rabs_right (/ INR (fact (N - n0)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) * Rabs x ^ S (n0 + n) *
(/ INR (fact (N - n0)) * Rabs y ^ (N - n0)) <=
M ^ (2 * N) * / INR (fact (S n0)) *
/ INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace
  (/ INR (fact (S (n0 + n))) * Rabs x ^ S (n0 + n) *
    (/ INR (fact (N - n0)) * Rabs y ^ (N - n0))) with
  (/ INR (fact (N - n0)) * / INR (fact (S (n0 + n))) * Rabs x ^ S (n0 + n) *
    Rabs y ^ (N - n0)); [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) * / INR (fact (S (n0 + n))) *
Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0) <=
M ^ (2 * N) * / INR (fact (S n0)) *
/ INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- (Rmult_comm (/ INR (fact (N - n0)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) * / INR (fact (S (n0 + n))) *
Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0) <=
/ INR (fact (N - n0)) *
(M ^ (2 * N) * / INR (fact (S n0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  repeat rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) *
(/ INR (fact (S (n0 + n))) *
 (Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0))) <=
/ INR (fact (N - n0)) *
(M ^ (2 * N) * / INR (fact (S n0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) *
(Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0)) <=
M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) *
(Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0)) <=
M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with
    (/ INR (fact (S n0)) * Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) *
(Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0)) <=
/ INR (fact (S n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rmult_comm (/ INR (fact (S (n0 + n)))));
    rewrite (Rmult_comm (/ INR (fact (S n0)))); repeat rewrite Rmult_assoc;
      apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ S (n0 + n)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0) * / INR (fact (S (n0 + n))) <=
/ INR (fact (S n0)) * Rabs y ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply pow_le; apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0) * / INR (fact (S (n0 + n))) <=
/ INR (fact (S n0)) * Rabs y ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rmult_comm (/ INR (fact (S n0)))); apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) <= / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply pow_le; apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) <= / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rinv_le_contravar.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) <= INR (fact (S (n0 + n)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) <= INR (fact (S (n0 + n)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_INR; apply fact_le; apply le_n_S.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= n0 + n)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_plus_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * Rabs x ^ S (n0 + n) *
Rabs y ^ (N - n0) <= M ^ (2 * N) * / INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rmult_comm (M ^ (2 * N))); rewrite Rmult_assoc;
    apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with (M ^ S (n0 + n) * Rabs y ^ (N - n0)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ S (n0 + n) * Rabs y ^ (N - n0) <=
M ^ S (n0 + n) * Rabs y ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  do 2 rewrite <- (Rmult_comm (Rabs y ^ (N - n0))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0) * Rabs x ^ S (n0 + n) <=
Rabs y ^ (N - n0) * M ^ S (n0 + n)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ S (n0 + n) <= M ^ S (n0 + n)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply pow_le; apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ S (n0 + n) <= M ^ S (n0 + n)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply pow_incr; split.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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) <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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) <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold M; apply RmaxLess2.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with (M ^ S (n0 + n) * M ^ (N - n0)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * Rabs y ^ (N - n0) <=
M ^ S (n0 + n) * M ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= M ^ S (n0 + n)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0) <= M ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply pow_le; apply Rle_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0) <= M ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  left; apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0) <= M ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold M; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 ^ (N - n0) <= M ^ (N - n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply pow_incr; split.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 (Rabs x) (Rabs y)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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) <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply RmaxLess2.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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) <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold M; apply RmaxLess2.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ S (n0 + n) * M ^ (N - n0) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- pow_add; replace (S (n0 + n) + (N - n0))%nat with (N + S n)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ (N + S n) <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 = (S (n0 + n) + (N - n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_pow.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= 2 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 = (S (n0 + n) + (N - n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold M; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= 2 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 = (S (n0 + n) + (N - n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= N + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 = (S (n0 + n) + (N - n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply plus_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 = (S (n0 + n) + (N - n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace N with (S (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= S (Init.Nat.pred N))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 = (S (n0 + n) + (N - n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_n_S; apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 = (S (n0 + n) + (N - n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  symmetry ; apply S_pred with 0%nat; apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 = (S (n0 + n) + (N - n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_eq; do 2 rewrite plus_INR; do 2 rewrite S_INR; rewrite plus_INR;
    rewrite minus_INR.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR N + (INR n + 1) =
INR n0 + INR n + 1 + (INR N - INR n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with (pred (N - n)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_S_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (S (pred (N - n))) with (N - n)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with N.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y 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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_plus_r.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply S_pred with 0%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply plus_lt_reg_l with n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (n + 0)%nat with n; [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_lt_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply lt_pred_n_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (N - n0)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite scal_sum.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      M ^ (2 * N) * / INR (fact (S l)) *
      / INR (fact (N - l))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
sum_f_R0
  (fun i : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - i)) * M ^ (2 * N))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   M ^ (2 * N) * / INR (fact (S l)) *
   / INR (fact (N - l))) (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred (N - n)) * M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- Rmult_comm.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   M ^ (2 * N) * / INR (fact (S l)) *
   / INR (fact (N - l))) (Init.Nat.pred (N - n)) <=
M ^ (2 * N) *
sum_f_R0
  (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite scal_sum.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   M ^ (2 * N) * / INR (fact (S l)) *
   / INR (fact (N - l))) (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun _ : nat =>
   / (INR (fact (Nat.div2 (S N))))² * M ^ (2 * N))
  (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ (2 * N) * / INR (fact (S n0)) *
/ INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))² * M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rmult_comm (/ Rsqr (INR (fact (div2 (S N)))))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
M ^ (2 * N) * / INR (fact (S n0)) *
/ INR (fact (N - n0)) <=
M ^ (2 * N) * / (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite Rmult_assoc; apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= M ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  left; apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (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 <= M
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold M; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (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 n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  assert (H2 := even_odd_cor N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  elim H2; intros N0 H3.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  elim H3; intro.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with (/ INR (fact n0) * / INR (fact (N - n0))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  do 2 rewrite <- (Rmult_comm (/ INR (fact (N - n0)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact (N - n0)) * / INR (fact (S n0)) <=
/ INR (fact (N - n0)) * / INR (fact n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
0 <= / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact (S n0)) <= / INR (fact n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact (S n0)) <= / INR (fact n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rinv_le_contravar.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
0 < INR (fact n0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact n0) <= INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact n0) <= INR (fact (S n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_INR.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(fact n0 <= fact (S n0))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply fact_le.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n0 <= S n0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (/ INR (fact n0) * / INR (fact (N - n0))) with
  (C N n0 / INR (fact N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  pattern N at 1; rewrite H4.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C (2 * N0) n0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with (C N N0 / INR (fact N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C (2 * N0) n0 / INR (fact N) <= C N N0 / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact N))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact N) * C (2 * N0) n0 <=
/ INR (fact N) * C N N0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
0 <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C (2 * N0) n0 <= C N N0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C (2 * N0) n0 <= C N N0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite H4.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C (2 * N0) n0 <= C (2 * N0) N0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply C_maj.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n0 <= 2 * N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- H4; apply le_trans with (pred (N - n)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n0 <= Init.Nat.pred (N - n))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(Init.Nat.pred (N - n) <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(Init.Nat.pred (N - n) <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_S_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(S (Init.Nat.pred (N - n)) <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (S (pred (N - n))) with (N - n)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with N.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y 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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n + (N - n) <= n + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N <= n + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_plus_r.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply S_pred with 0%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(0 < N - n)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply plus_lt_reg_l with n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n + 0 < n + (N - n))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n + 0 < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (n + 0)%nat with n; [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_lt_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(Init.Nat.pred N < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(Init.Nat.pred N < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply lt_pred_n_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(0 < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N N0 / INR (fact N) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (C N N0 / INR (fact N)) with (/ Rsqr (INR (fact N0))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ (INR (fact N0))² <= / (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ (INR (fact N0))² = C N N0 / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite H4; rewrite div2_S_double; right; reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ (INR (fact N0))² = C N N0 / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold Rsqr, C, Rdiv.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ (INR (fact N0) * INR (fact N0)) =
INR (fact N) * / (INR (fact N0) * INR (fact (N - N0))) *
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  repeat rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact N0) * / INR (fact N0) =
INR (fact N) *
(/ INR (fact N0) * / INR (fact (N - N0))) *
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rmult_comm (INR (fact N))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact N0) * / INR (fact N0) =
/ INR (fact N0) * / INR (fact (N - N0)) * INR (fact N) *
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  repeat rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact N0) * / INR (fact N0) =
/ INR (fact N0) *
(/ INR (fact (N - N0)) *
 (INR (fact N) * / INR (fact N)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact N0) * / INR (fact N0) =
/ INR (fact N0) * (/ INR (fact (N - N0)) * 1)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite Rmult_1_r; replace (N - N0)%nat with N0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact N0) * / INR (fact N0) =
/ INR (fact N0) * / INR (fact N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
N0 = (N - N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
N0 = (N - N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace N with (N0 + N0)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
N0 = (N0 + N0 - N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N0 + N0)%nat = N
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  symmetry ; apply minus_plus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N0 + N0)%nat = N
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite H4.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
(N0 + N0)%nat = (2 * N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
C N n0 / INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold C, Rdiv.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) * / (INR (fact n0) * INR (fact (N - n0))) *
/ INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rmult_comm (INR (fact N))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ (INR (fact n0) * INR (fact (N - n0))) * INR (fact N) *
/ INR (fact N) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  repeat rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ (INR (fact n0) * INR (fact (N - n0))) *
(INR (fact N) * / INR (fact N)) =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ (INR (fact n0) * INR (fact (N - n0))) * 1 =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
/ INR (fact n0) * / INR (fact (N - n0)) * 1 =
/ INR (fact n0) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact n0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - n0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite Rmult_1_r; ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact n0) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - n0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact (N - n0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = (2 * N0)%nat
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (N - n0)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (/ INR (fact (S n0)) * / INR (fact (N - n0))) with
  (C (S N) (S n0) / INR (fact (S N))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rle_trans with (C (S N) (S N0) / INR (fact (S N))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) <=
C (S N) (S N0) / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact (S N)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S N)) * C (S N) (S n0) <=
/ INR (fact (S N)) * C (S N) (S N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
0 <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) <= C (S N) (S N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) <= C (S N) (S N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  cut (S N = (2 * S N0)%nat).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat ->
C (S N) (S n0) <= C (S N) (S N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  intro; rewrite H5; apply C_maj.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(S n0 <= 2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- H5; apply le_n_S.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n0 <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with (pred (N - n)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n0 <= Init.Nat.pred (N - n))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(Init.Nat.pred (N - n) <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(Init.Nat.pred (N - n) <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_S_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(S (Init.Nat.pred (N - n)) <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (S (pred (N - n))) with (N - n)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with N.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y 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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n + (N - n) <= n + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N <= n + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_plus_r.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply S_pred with 0%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(0 < N - n)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply plus_lt_reg_l with n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n + 0 < n + (N - n))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n + 0 < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (n + 0)%nat with n; [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_lt_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(Init.Nat.pred N < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(Init.Nat.pred N < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply lt_pred_n_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(0 < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite H4; ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  cut (S N = (2 * S N0)%nat).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat ->
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  intro.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
C (S N) (S N0) / INR (fact (S N)) <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (C (S N) (S N0) / INR (fact (S N))) with (/ Rsqr (INR (fact (S N0)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ (INR (fact (S N0)))² <=
/ (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ (INR (fact (S N0)))² =
C (S N) (S N0) / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite H5; rewrite div2_double.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ (INR (fact (S N0)))² <= / (INR (fact (S N0)))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ (INR (fact (S N0)))² =
C (S N) (S N0) / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  right; reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ (INR (fact (S N0)))² =
C (S N) (S N0) / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold Rsqr, C, Rdiv.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ (INR (fact (S N0)) * INR (fact (S N0))) =
INR (fact (S N)) *
/ (INR (fact (S N0)) * INR (fact (S N - S N0))) *
/ INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  repeat rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ INR (fact (S N0)) * / INR (fact (S N0)) =
INR (fact (S N)) *
(/ INR (fact (S N0)) * / INR (fact (S N - S N0))) *
/ INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (S N - S N0)%nat with (S N0).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ INR (fact (S N0)) * / INR (fact (S N0)) =
INR (fact (S N)) *
(/ INR (fact (S N0)) * / INR (fact (S N0))) *
/ INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
S N0 = (S N - S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rmult_comm (INR (fact (S N)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ INR (fact (S N0)) * / INR (fact (S N0)) =
/ INR (fact (S N0)) * / INR (fact (S N0)) *
INR (fact (S N)) * / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
S N0 = (S N - S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  repeat rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ INR (fact (S N0)) * / INR (fact (S N0)) =
/ INR (fact (S N0)) *
(/ INR (fact (S N0)) *
 (INR (fact (S N)) * / INR (fact (S N))))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
S N0 = (S N - S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
/ INR (fact (S N0)) * / INR (fact (S N0)) =
/ INR (fact (S N0)) * (/ INR (fact (S N0)) * 1)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
S N0 = (S N - S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite Rmult_1_r; reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
S N0 = (S N - S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
S N0 = (S N - S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  replace (S N) with (S N0 + S N0)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
S N0 = (S N0 + S N0 - S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(S N0 + S N0)%nat = S N
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  symmetry ; apply minus_plus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
(S N0 + S N0)%nat = S N
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite H5; ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N - S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
H5:S N = (2 * S N0)%nat
INR (fact (S N0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
S N = (2 * S N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite H4; ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
C (S N) (S n0) / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold C, Rdiv.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N)) *
/ (INR (fact (S n0)) * INR (fact (S N - S n0))) *
/ INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite (Rmult_comm (INR (fact (S N)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ (INR (fact (S n0)) * INR (fact (S N - S n0))) *
INR (fact (S N)) * / INR (fact (S N)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ (INR (fact (S n0)) * INR (fact (S N - S n0))) * 1 =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  rewrite Rmult_1_r; rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
/ INR (fact (S n0)) * / INR (fact (S N - S n0)) =
/ INR (fact (S n0)) * / INR (fact (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S n0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N - S n0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S n0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N - S n0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N - S n0)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
H2:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H3:N = (2 * N0)%nat \/ N = S (2 * N0)
H4:N = S (2 * N0)
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
maj_Reste_E x y N
  unfold maj_Reste_E.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * N) /
 (INR (fact (Nat.div2 (Init.Nat.pred N))))²)
  unfold Rdiv; rewrite (Rmult_comm 4).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * N) *
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
M ^ (2 * N) *
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * N) *
(/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 <= Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply Rle_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 <= 1
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
1 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  left; apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
1 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply Rle_trans with
    (sum_f_R0 (fun k:nat => INR (N - k) * / Rsqr (INR (fact (div2 (S N)))))
      (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun _ : nat => / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred (N - n)) <=
INR (N - n) * / (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  rewrite sum_cte.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ (INR (fact (Nat.div2 (S N))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR (N - n) * / (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  replace (S (pred (N - n))) with (N - n)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ (INR (fact (Nat.div2 (S N))))² * INR (N - n) <=
INR (N - n) * / (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  right; apply Rmult_comm.
x, y:R
N:nat
H:(0 < N)%nat
M:=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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply S_pred with 0%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(0 < N - n)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply plus_lt_reg_l with n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n + 0 < n + (N - n))%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n + 0 < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  replace (n + 0)%nat with n; [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply le_lt_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(Init.Nat.pred N < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(Init.Nat.pred N < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply lt_pred_n_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(0 < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply Rle_trans with
    (sum_f_R0 (fun k:nat => INR N * / Rsqr (INR (fact (div2 (S N))))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   INR (N - k) * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (N - n) * / (INR (fact (Nat.div2 (S N))))² <=
INR N * / (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  do 2 rewrite <- (Rmult_comm (/ Rsqr (INR (fact (div2 (S N)))))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ (INR (fact (Nat.div2 (S N))))² * INR (N - n) <=
/ (INR (fact (Nat.div2 (S N))))² * INR N
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= / (INR (fact (Nat.div2 (S N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (Nat.div2 (S N))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply le_INR.
x, y:R
N:nat
H:(0 < N)%nat
M:=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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply (fun p n m:nat => plus_le_reg_l n m p) with n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n + (N - n) <= n + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
M:=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
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply le_plus_r.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N * / (INR (fact (Nat.div2 (S N))))²)
  (Init.Nat.pred N) <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
  rewrite sum_cte; replace (S (pred N)) with N.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / (INR (fact (Nat.div2 (S N))))² * INR N <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  cut (div2 (S N) = S (div2 (pred N))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N)) ->
INR N * / (INR (fact (Nat.div2 (S N))))² * INR N <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  intro; rewrite H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N *
/ (INR (fact (S (Nat.div2 (Init.Nat.pred N)))))² *
INR N <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite fact_simpl; rewrite mult_comm; rewrite mult_INR; rewrite Rsqr_mult.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N *
/
((INR (fact (Nat.div2 (Init.Nat.pred N))))² *
 (INR (S (Nat.div2 (Init.Nat.pred N))))²) * INR N <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N *
(/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² *
 / (INR (S (Nat.div2 (Init.Nat.pred N))))²) * INR N <=
/ (INR (fact (Nat.div2 (Init.Nat.pred N))))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite (Rmult_comm (INR N)); repeat rewrite Rmult_assoc;
    apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
0 <= / (INR (fact (Nat.div2 (Init.Nat.pred N))))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
/ (INR (S (Nat.div2 (Init.Nat.pred N))))² *
(INR N * INR N) <= 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
/ (INR (S (Nat.div2 (Init.Nat.pred N))))² *
(INR N * INR N) <= 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite <- H0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
/ (INR (Nat.div2 (S N)))² * (INR N * INR N) <= 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  cut (INR N <= INR (2 * div2 (S N))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N)) ->
/ (INR (Nat.div2 (S N)))² * (INR N * INR N) <= 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  intro; apply Rmult_le_reg_l with (Rsqr (INR (div2 (S N)))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 < (INR (Nat.div2 (S N)))²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² *
(/ (INR (Nat.div2 (S N)))² * (INR N * INR N)) <=
(INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply Rsqr_pos_lt.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
INR (Nat.div2 (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² *
(/ (INR (Nat.div2 (S N)))² * (INR N * INR N)) <=
(INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply not_O_INR; red; intro.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
H2:Nat.div2 (S N) = 0%nat
False
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² *
(/ (INR (Nat.div2 (S N)))² * (INR N * INR N)) <=
(INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  cut (1 < S N)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
H2:Nat.div2 (S N) = 0%nat
(1 < S N)%nat -> False
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
H2:Nat.div2 (S N) = 0%nat
(1 < S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² *
(/ (INR (Nat.div2 (S N)))² * (INR N * INR N)) <=
(INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  intro; assert (H4 := div2_not_R0 _ H3).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
H2:Nat.div2 (S N) = 0%nat
H3:(1 < S N)%nat
H4:(0 < Nat.div2 (S N))%nat
False
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
H2:Nat.div2 (S N) = 0%nat
(1 < S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² *
(/ (INR (Nat.div2 (S N)))² * (INR N * INR N)) <=
(INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite H2 in H4; elim (lt_n_O _ H4).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
H2:Nat.div2 (S N) = 0%nat
(1 < S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² *
(/ (INR (Nat.div2 (S N)))² * (INR N * INR N)) <=
(INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply lt_n_S; apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² *
(/ (INR (Nat.div2 (S N)))² * (INR N * INR N)) <=
(INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  repeat rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² * / (INR (Nat.div2 (S N)))² *
INR N * INR N <= (INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
1 * INR N * INR N <= (INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
INR N * INR N <= (INR (Nat.div2 (S N)))² * 4
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  change 4 with (Rsqr 2).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
INR N * INR N <= (INR (Nat.div2 (S N)))² * 2²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite <- Rsqr_mult.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
INR N * INR N <= (INR (Nat.div2 (S N)) * 2)²
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply Rsqr_incr_1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
INR N <= INR (Nat.div2 (S N)) * 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 <= INR (Nat.div2 (S N)) * 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  change 2 with (INR 2).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
INR N <= INR (Nat.div2 (S N)) * INR 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 <= INR (Nat.div2 (S N)) * 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite Rmult_comm, <- mult_INR; apply H1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 <= INR (Nat.div2 (S N)) * 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  left; apply lt_INR_0; apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 <= INR (Nat.div2 (S N)) * 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  left; apply Rmult_lt_0_compat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 < INR (Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 < 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply lt_INR_0; apply div2_not_R0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(1 < S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 < 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply lt_n_S; apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
0 < 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  now apply IZR_lt.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  cut (1 < S N)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(1 < S N)%nat -> (INR (Nat.div2 (S N)))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(1 < S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  intro; unfold Rsqr; apply prod_neq_R0; apply not_O_INR; intro;
    assert (H4 := div2_not_R0 _ H2); rewrite H3 in H4;
      elim (lt_n_O _ H4).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:INR N <= INR (2 * Nat.div2 (S N))
(1 < S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply lt_n_S; apply H.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  assert (H1 := even_odd_cor N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  elim H1; intros N0 H2.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  elim H2; intro.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = (2 * N0)%nat
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  pattern N at 2; rewrite H3.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = (2 * N0)%nat
INR N <= INR (2 * Nat.div2 (S (2 * N0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite div2_S_double.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = (2 * N0)%nat
INR N <= INR (2 * N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  right; rewrite H3; reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR N <= INR (2 * Nat.div2 (S N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  pattern N at 2; rewrite H3.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR N <= INR (2 * Nat.div2 (S (S (2 * N0))))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  replace (S (S (2 * N0))) with (2 * S N0)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR N <= INR (2 * Nat.div2 (2 * S N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite div2_double.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR N <= INR (2 * S N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite H3.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR (S (2 * N0)) <= INR (2 * S N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite S_INR; do 2 rewrite mult_INR.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR 2 * INR N0 + 1 <= INR 2 * INR (S N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite (S_INR N0).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR 2 * INR N0 + 1 <= INR 2 * (INR N0 + 1)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite Rmult_plus_distr_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
INR 2 * INR N0 + 1 <= INR 2 * INR N0 + INR 2 * 1
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply Rplus_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
1 <= INR 2 * 1
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
1 <= INR 2
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  simpl.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
1 <= 1 + 1
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
    apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
H1:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H2:N = (2 * N0)%nat \/ N = S (2 * N0)
H3:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (fact (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  unfold Rsqr; apply prod_neq_R0; apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
(INR (S (Nat.div2 (Init.Nat.pred N))))² <> 0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  unfold Rsqr; apply prod_neq_R0; apply not_O_INR; discriminate.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  assert (H0 := even_odd_cor N).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  elim H0; intros N0 H1.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  elim H1; intro.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  cut (0 < N0)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat ->
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  intro; rewrite H2.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
Nat.div2 (S (2 * N0)) =
S (Nat.div2 (Init.Nat.pred (2 * N0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite div2_S_double.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
N0 = S (Nat.div2 (Init.Nat.pred (2 * N0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  replace (2 * N0)%nat with (S (S (2 * pred N0))).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
N0 =
S
  (Nat.div2
     (Init.Nat.pred (S (S (2 * Init.Nat.pred N0)))))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
S (S (2 * Init.Nat.pred N0)) = (2 * N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  replace (pred (S (S (2 * pred N0)))) with (S (2 * pred N0)).
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
N0 = S (Nat.div2 (S (2 * Init.Nat.pred N0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
S (2 * Init.Nat.pred N0) =
Init.Nat.pred (S (S (2 * Init.Nat.pred N0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
S (S (2 * Init.Nat.pred N0)) = (2 * N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite div2_S_double.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
N0 = S (Init.Nat.pred N0)
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
S (2 * Init.Nat.pred N0) =
Init.Nat.pred (S (S (2 * Init.Nat.pred N0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
S (S (2 * Init.Nat.pred N0)) = (2 * N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply S_pred with 0%nat; apply H3.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
S (2 * Init.Nat.pred N0) =
Init.Nat.pred (S (S (2 * Init.Nat.pred N0)))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
S (S (2 * Init.Nat.pred N0)) = (2 * N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
H3:(0 < N0)%nat
S (S (2 * Init.Nat.pred N0)) = (2 * N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  omega.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = (2 * N0)%nat
(0 < N0)%nat
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  omega.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S N) = S (Nat.div2 (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  rewrite H2.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S (S (2 * N0))) =
S (Nat.div2 (Init.Nat.pred (S (2 * N0))))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  replace (pred (S (2 * N0))) with (2 * N0)%nat; [ idtac | reflexivity ].
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (S (S (2 * N0))) = S (Nat.div2 (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  replace (S (S (2 * N0))) with (2 * S N0)%nat.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
Nat.div2 (2 * S N0) = S (Nat.div2 (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  do 2 rewrite div2_double.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
S N0 = S N0
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:exists p : nat, N = (2 * p)%nat \/ N = S (2 * p)
N0:nat
H1:N = (2 * N0)%nat \/ N = S (2 * N0)
H2:N = S (2 * N0)
(2 * S N0)%nat = S (S (2 * N0))
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  ring.
x, y:R
N:nat
H:(0 < N)%nat
M:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply S_pred with 0%nat; apply H.
Qed.

Lemma maj_Reste_cv_R0 : forall x y:R, Un_cv (maj_Reste_E x y) 0.
forall x y : R, Un_cv (maj_Reste_E x y) 0
Proof.
forall x y : R, Un_cv (maj_Reste_E x y) 0
  intros; assert (H := Majxy_cv_R0 x y).
x, y:R
H:Un_cv (Majxy x y) 0
Un_cv (maj_Reste_E x y) 0
  unfold Un_cv in H; unfold Un_cv; intros.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Majxy x y n) 0 < eps0
eps:R
H0:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (maj_Reste_E x y n) 0 < eps
  cut (0 < eps / 4);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
        [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Majxy x y n) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (maj_Reste_E x y n) 0 < eps
  elim (H _ H1); intros N0 H2.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Majxy x y n) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n : nat,
(n >= N0)%nat -> R_dist (Majxy x y n) 0 < eps / 4
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (maj_Reste_E x y n) 0 < eps
  exists (max (2 * S N0) 2); intros.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist (Majxy x y n0) 0 < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
R_dist (maj_Reste_E x y n) 0 < eps
  unfold R_dist in H2; unfold R_dist; rewrite Rminus_0_r;
    unfold Majxy in H2; unfold maj_Reste_E.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (4 *
   (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
    (INR (fact (Nat.div2 (Init.Nat.pred n))))²)) < eps
  rewrite Rabs_right.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rle_lt_trans with
    (4 *
      (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) /
        INR (fact (div2 (pred n))))).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) <=
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n))))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rmult_le_compat_l.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= 4
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
(INR (fact (Nat.div2 (Init.Nat.pred n))))² <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  left; prove_sup0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
(INR (fact (Nat.div2 (Init.Nat.pred n))))² <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) *
(/ INR (fact (Nat.div2 (Init.Nat.pred n))) *
 / INR (fact (Nat.div2 (Init.Nat.pred n)))) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) *
/ INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)));
    rewrite
      (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n)))))
      ; rewrite Rmult_assoc; apply Rmult_le_compat_l.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ INR (fact (Nat.div2 (Init.Nat.pred n))) *
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ INR (fact (Nat.div2 (Init.Nat.pred n))) *
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rle_trans with (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ INR (fact (Nat.div2 (Init.Nat.pred n))) *
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite Rmult_comm;
    pattern (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)) at 2;
      rewrite <- Rmult_1_r; apply Rmult_le_compat_l.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ INR (fact (Nat.div2 (Init.Nat.pred n))) <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply pow_le; apply Rle_trans with 1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ INR (fact (Nat.div2 (Init.Nat.pred n))) <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  left; apply Rlt_0_1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ INR (fact (Nat.div2 (Init.Nat.pred n))) <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply RmaxLess1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ INR (fact (Nat.div2 (Init.Nat.pred n))) <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rmult_le_reg_l with (INR (fact (div2 (pred n)))).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 < INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) *
/ INR (fact (Nat.div2 (Init.Nat.pred n))) <=
INR (fact (Nat.div2 (Init.Nat.pred n))) * 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply INR_fact_lt_0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) *
/ INR (fact (Nat.div2 (Init.Nat.pred n))) <=
INR (fact (Nat.div2 (Init.Nat.pred n))) * 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite Rmult_1_r; rewrite <- Rinv_r_sym.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
1 <= INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply (le_INR 1).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(1 <= fact (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply lt_le_S.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(0 < fact (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply INR_lt.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR 0 < INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply INR_fact_lt_0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply INR_fact_neq_0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rle_pow.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply RmaxLess1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  assert (H4 := even_odd_cor n).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  elim H4; intros N1 H5.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  elim H5; intro.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  cut (0 < N1)%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < N1)%nat ->
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  intro.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < N1)%nat
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite H6.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < N1)%nat
(2 * (2 * N1) <=
 4 * S (Nat.div2 (Init.Nat.pred (2 * N1))))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (pred (2 * N1)) with (S (2 * pred N1)).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < N1)%nat
(2 * (2 * N1) <=
 4 * S (Nat.div2 (S (2 * Init.Nat.pred N1))))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < N1)%nat
S (2 * Init.Nat.pred N1) = Init.Nat.pred (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite div2_S_double.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < N1)%nat
(2 * (2 * N1) <= 4 * S (Init.Nat.pred N1))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < N1)%nat
S (2 * Init.Nat.pred N1) = Init.Nat.pred (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  omega.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < N1)%nat
S (2 * Init.Nat.pred N1) = Init.Nat.pred (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  omega.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  assert (0 < n)%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < n)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply lt_le_trans with 2%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(0 < 2)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < n)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply lt_O_Sn.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < n)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_trans with (max (2 * S N0) 2).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(2 <= Nat.max (2 * S N0) 2)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(Nat.max (2 * S N0) 2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < n)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_max_r.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
(Nat.max (2 * S N0) 2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < n)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply H3.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = (2 * N1)%nat
H7:(0 < n)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  omega.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * n <= 4 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite H6.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * S (2 * N1) <=
 4 * S (Nat.div2 (Init.Nat.pred (S (2 * N1)))))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (pred (S (2 * N1))) with (2 * N1)%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * S (2 * N1) <= 4 * S (Nat.div2 (2 * N1)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite div2_double.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * S (2 * N1) <= 4 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (4 * S N1)%nat with (2 * (2 * S N1))%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * S (2 * N1) <= 2 * (2 * S N1))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * (2 * S N1))%nat = (4 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply (fun m n p:nat => mult_le_compat_l p n m).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(S (2 * N1) <= 2 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * (2 * S N1))%nat = (4 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (2 * S N1)%nat with (S (S (2 * N1))).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(S (2 * N1) <= S (S (2 * N1)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
S (S (2 * N1)) = (2 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * (2 * S N1))%nat = (4 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_n_Sn.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
S (S (2 * N1)) = (2 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * (2 * S N1))%nat = (4 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  ring.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * (2 * S N1))%nat = (4 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  ring.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H5:n = (2 * N1)%nat \/ n = S (2 * N1)
H6:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  reflexivity.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply INR_fact_neq_0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
INR (fact (Nat.div2 (Init.Nat.pred n))) <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply INR_fact_neq_0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rmult_lt_reg_l with (/ 4).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 < / 4
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ 4 *
(4 *
 (Rmax 1 (Rmax (Rabs x) (Rabs y))
  ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
  INR (fact (Nat.div2 (Init.Nat.pred n))))) <
/ 4 * eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rinv_0_lt_compat; prove_sup0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
/ 4 *
(4 *
 (Rmax 1 (Rmax (Rabs x) (Rabs y))
  ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
  INR (fact (Nat.div2 (Init.Nat.pred n))))) <
/ 4 * eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
1 *
(Rmax 1 (Rmax (Rabs x) (Rabs y))
 ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
 INR (fact (Nat.div2 (Init.Nat.pred n)))) < / 4 * eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite Rmult_1_l; rewrite Rmult_comm.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n))) < eps * / 4
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) /
    INR (fact (div2 (pred n)))) with
  (Rabs
    (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) /
      INR (fact (div2 (pred n))) - 0)).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) <
eps * / 4
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply H2; unfold ge.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(N0 <= Nat.div2 (Init.Nat.pred n))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  cut (2 * S N0 <= n)%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat ->
(N0 <= Nat.div2 (Init.Nat.pred n))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  intro; apply le_S_n.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
(S N0 <= S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply INR_le; apply Rmult_le_reg_l with (INR 2).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
0 < INR 2
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
INR 2 * INR (S N0) <=
INR 2 * INR (S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  simpl; prove_sup0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
INR 2 * INR (S N0) <=
INR 2 * INR (S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  do 2 rewrite <- mult_INR; apply le_INR.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
(2 * S N0 <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_trans with n.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply H4.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  assert (H5 := even_odd_cor n).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  elim H5; intros N1 H6.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  elim H6; intro.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  cut (0 < N1)%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat ->
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  intro.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite H7.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
(2 * N1 <= 2 * S (Nat.div2 (Init.Nat.pred (2 * N1))))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply (fun m n p:nat => mult_le_compat_l p n m).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
(N1 <= S (Nat.div2 (Init.Nat.pred (2 * N1))))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (pred (2 * N1)) with (S (2 * pred N1)).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
(N1 <= S (Nat.div2 (S (2 * Init.Nat.pred N1))))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (2 * Init.Nat.pred N1) = Init.Nat.pred (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite div2_S_double.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
(N1 <= S (Init.Nat.pred N1))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (2 * Init.Nat.pred N1) = Init.Nat.pred (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (S (pred N1)) with N1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
(N1 <= N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
N1 = S (Init.Nat.pred N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (2 * Init.Nat.pred N1) = Init.Nat.pred (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_n.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
N1 = S (Init.Nat.pred N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (2 * Init.Nat.pred N1) = Init.Nat.pred (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply S_pred with 0%nat; apply H8.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (2 * Init.Nat.pred N1) = Init.Nat.pred (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (2 * N1)%nat with (S (S (2 * pred N1))).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (2 * Init.Nat.pred N1) =
Init.Nat.pred (S (S (2 * Init.Nat.pred N1)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (S (2 * Init.Nat.pred N1)) = (2 * N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  reflexivity.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (S (2 * Init.Nat.pred N1)) = (2 * N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  pattern N1 at 2; replace N1 with (S (pred N1)).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (S (2 * Init.Nat.pred N1)) =
(2 * S (Init.Nat.pred N1))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (Init.Nat.pred N1) = N1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  ring.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
H8:(0 < N1)%nat
S (Init.Nat.pred N1) = N1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  symmetry ; apply S_pred with 0%nat; apply H8.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply INR_lt.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
INR 0 < INR N1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rmult_lt_reg_l with (INR 2).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
0 < INR 2
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
INR 2 * INR 0 < INR 2 * INR N1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  simpl; prove_sup0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
INR 2 * INR 0 < INR 2 * INR N1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite Rmult_0_r; rewrite <- mult_INR.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
0 < INR (2 * N1)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply lt_INR_0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < 2 * N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite <- H7.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply lt_le_trans with 2%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(0 < 2)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply lt_O_Sn.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_trans with (max (2 * S N0) 2).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(2 <= Nat.max (2 * S N0) 2)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(Nat.max (2 * S N0) 2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_max_r.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = (2 * N1)%nat
(Nat.max (2 * S N0) 2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply H3.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(n <= 2 * S (Nat.div2 (Init.Nat.pred n)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite H7.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(S (2 * N1) <=
 2 * S (Nat.div2 (Init.Nat.pred (S (2 * N1)))))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (pred (S (2 * N1))) with (2 * N1)%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(S (2 * N1) <= 2 * S (Nat.div2 (2 * N1)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite div2_double.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(S (2 * N1) <= 2 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  replace (2 * S N1)%nat with (S (S (2 * N1))).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(S (2 * N1) <= S (S (2 * N1)))%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
S (S (2 * N1)) = (2 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_n_Sn.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
S (S (2 * N1)) = (2 * S N1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  ring.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
H4:(2 * S N0 <= n)%nat
H5:exists p : nat, n = (2 * p)%nat \/ n = S (2 * p)
N1:nat
H6:n = (2 * N1)%nat \/ n = S (2 * N1)
H7:n = S (2 * N1)
(2 * N1)%nat = Init.Nat.pred (S (2 * N1))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  reflexivity.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_trans with (max (2 * S N0) 2).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(2 * S N0 <= Nat.max (2 * S N0) 2)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(Nat.max (2 * S N0) 2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply le_max_l.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
(Nat.max (2 * S N0) 2 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply H3.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y))
   ^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
   INR (fact (Nat.div2 (Init.Nat.pred n))) - 0) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  rewrite Rminus_0_r; apply Rabs_right.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n))) >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rle_ge.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n))) /
INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  unfold Rdiv; apply Rmult_le_pos.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply pow_le.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rle_trans with 1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  left; apply Rlt_0_1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply RmaxLess1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / INR (fact (Nat.div2 (Init.Nat.pred n)))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 <> 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  discrR.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²) >= 0
  apply Rle_ge.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <=
4 *
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) /
 (INR (fact (Nat.div2 (Init.Nat.pred n))))²)
  unfold Rdiv; apply Rmult_le_pos.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= 4
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) *
/ (INR (fact (Nat.div2 (Init.Nat.pred n))))²
  left; prove_sup0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n) *
/ (INR (fact (Nat.div2 (Init.Nat.pred n))))²
  apply Rmult_le_pos.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / (INR (fact (Nat.div2 (Init.Nat.pred n))))²
  apply pow_le.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / (INR (fact (Nat.div2 (Init.Nat.pred n))))²
  apply Rle_trans with 1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / (INR (fact (Nat.div2 (Init.Nat.pred n))))²
  left; apply Rlt_0_1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%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 -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / (INR (fact (Nat.div2 (Init.Nat.pred n))))²
  apply RmaxLess1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (Majxy x y n0) 0 < eps0
eps:R
H0:eps > 0
H1:0 < eps / 4
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n0) /
   INR (fact n0) - 0) < eps / 4
n:nat
H3:(n >= Nat.max (2 * S N0) 2)%nat
0 <= / (INR (fact (Nat.div2 (Init.Nat.pred n))))²
  left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; apply INR_fact_neq_0.
Qed.

(**********)
Lemma Reste_E_cv : forall x y:R, Un_cv (Reste_E x y) 0.
forall x y : R, Un_cv (Reste_E x y) 0
Proof.
forall x y : R, Un_cv (Reste_E x y) 0
  intros; assert (H := maj_Reste_cv_R0 x y).
x, y:R
H:Un_cv (maj_Reste_E x y) 0
Un_cv (Reste_E x y) 0
  unfold Un_cv in H; unfold Un_cv; intros; elim (H _ H0); intros.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (maj_Reste_E x y n) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n : nat,
(n >= x0)%nat ->
R_dist (maj_Reste_E x y n) 0 < eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Reste_E x y n) 0 < eps
  exists (max x0 1); intros.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
R_dist (Reste_E x y n) 0 < eps
  unfold R_dist; rewrite Rminus_0_r.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
Rabs (Reste_E x y n) < eps
  apply Rle_lt_trans with (maj_Reste_E x y n).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
Rabs (Reste_E x y n) <= maj_Reste_E x y n
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
maj_Reste_E x y n < eps
  apply Reste_E_maj.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(0 < n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
maj_Reste_E x y n < eps
  apply lt_le_trans with 1%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(0 < 1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(1 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
maj_Reste_E 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 ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(1 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
maj_Reste_E x y n < eps
  apply le_trans with (max x0 1).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(1 <= Nat.max x0 1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(Nat.max x0 1 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
maj_Reste_E x y n < eps
  apply le_max_r.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(Nat.max x0 1 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
maj_Reste_E x y n < eps
  apply H2.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
maj_Reste_E x y n < eps
  replace (maj_Reste_E x y n) with (R_dist (maj_Reste_E x y n) 0).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
R_dist (maj_Reste_E x y n) 0 < eps
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
R_dist (maj_Reste_E x y n) 0 = maj_Reste_E x y n
  apply H1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(n >= x0)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
R_dist (maj_Reste_E x y n) 0 = maj_Reste_E x y n
  unfold ge; apply le_trans with (max x0 1).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(x0 <= Nat.max x0 1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(Nat.max x0 1 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
R_dist (maj_Reste_E x y n) 0 = maj_Reste_E x y n
  apply le_max_l.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(Nat.max x0 1 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
R_dist (maj_Reste_E x y n) 0 = maj_Reste_E x y n
  apply H2.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
R_dist (maj_Reste_E x y n) 0 = maj_Reste_E x y n
  unfold R_dist; rewrite Rminus_0_r; apply Rabs_right.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
maj_Reste_E x y n >= 0
  apply Rle_ge; apply Rle_trans with (Rabs (Reste_E x y n)).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
0 <= Rabs (Reste_E x y n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
Rabs (Reste_E x y n) <= maj_Reste_E x y n
  apply Rabs_pos.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
Rabs (Reste_E x y n) <= maj_Reste_E x y n
  apply Reste_E_maj.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(0 < n)%nat
  apply lt_le_trans with 1%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(0 < 1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(1 <= n)%nat
  apply lt_O_Sn.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(1 <= n)%nat
  apply le_trans with (max x0 1).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(1 <= Nat.max x0 1)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(Nat.max x0 1 <= n)%nat
  apply le_max_r.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (maj_Reste_E x y n0) 0 < eps0
eps:R
H0:eps > 0
x0:nat
H1:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (maj_Reste_E x y n0) 0 < eps
n:nat
H2:(n >= Nat.max x0 1)%nat
(Nat.max x0 1 <= n)%nat
  apply H2.
Qed.

(**********)
Lemma exp_plus : forall x y:R, exp (x + y) = exp x * exp y.
forall x y : R, exp (x + y) = exp x * exp y
Proof.
forall x y : R, exp (x + y) = exp x * exp y
  intros; assert (H0 := E1_cvg x).
x, y:R
H0:Un_cv (E1 x) (exp x)
exp (x + y) = exp x * exp y
  assert (H := E1_cvg y).
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
exp (x + y) = exp x * exp y
  assert (H1 := E1_cvg (x + y)).
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
exp (x + y) = exp x * exp y
  eapply UL_sequence.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
Un_cv ?Un (exp (x + y))
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
Un_cv ?Un (exp x * exp y)
  apply H1.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
Un_cv (E1 (x + y)) (exp x * exp y)
  assert (H2 := CV_mult _ _ _ _ H0 H).
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
Un_cv (E1 (x + y)) (exp x * exp y)
  assert (H3 := CV_minus _ _ _ _ H2 (Reste_E_cv x y)).
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:Un_cv
  (fun i : nat =>
   (fun i0 : nat => E1 x i0 * E1 y i0) i -
   Reste_E x y i) (exp x * exp y - 0)
Un_cv (E1 (x + y)) (exp x * exp y)
  unfold Un_cv; unfold Un_cv in H3; intros.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (E1 x n * E1 y n - Reste_E x y n)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (E1 (x + y) n) (exp x * exp y) < eps
  elim (H3 _ H4); intros.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (E1 x n * E1 y n - Reste_E x y n)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n : nat,
(n >= x0)%nat ->
R_dist (E1 x n * E1 y n - Reste_E x y n)
  (exp x * exp y - 0) < eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (E1 (x + y) n) (exp x * exp y) < eps
  exists (S x0); intros.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
R_dist (E1 (x + y) n) (exp x * exp y) < eps
  rewrite <- (exp_form x y n).
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
R_dist (E1 x n * E1 y n - Reste_E x y n)
  (exp x * exp y) < eps
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(0 < n)%nat
  rewrite Rminus_0_r in H5.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y) < eps
n:nat
H6:(n >= S x0)%nat
R_dist (E1 x n * E1 y n - Reste_E x y n)
  (exp x * exp y) < eps
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(0 < n)%nat
  apply H5.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y) < eps
n:nat
H6:(n >= S x0)%nat
(n >= x0)%nat
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(0 < n)%nat
  unfold ge; apply le_trans with (S x0).
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y) < eps
n:nat
H6:(n >= S x0)%nat
(x0 <= S x0)%nat
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y) < eps
n:nat
H6:(n >= S x0)%nat
(S x0 <= n)%nat
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(0 < n)%nat
  apply le_n_Sn.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y) < eps
n:nat
H6:(n >= S x0)%nat
(S x0 <= n)%nat
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(0 < n)%nat
  apply H6.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(0 < n)%nat
  apply lt_le_trans with (S x0).
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(0 < S x0)%nat
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(S x0 <= n)%nat
  apply lt_O_Sn.
x, y:R
H0:Un_cv (E1 x) (exp x)
H:Un_cv (E1 y) (exp y)
H1:Un_cv (E1 (x + y)) (exp (x + y))
H2:Un_cv (fun i : nat => E1 x i * E1 y i)
  (exp x * exp y)
H3:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
    (exp x * exp y - 0) < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (E1 x n0 * E1 y n0 - Reste_E x y n0)
  (exp x * exp y - 0) < eps
n:nat
H6:(n >= S x0)%nat
(S x0 <= n)%nat
  apply H6.
Qed.

(**********)
Lemma exp_pos_pos : forall x:R, 0 < x -> 0 < exp x.
forall x : R, 0 < x -> 0 < exp x
Proof.
forall x : R, 0 < x -> 0 < exp x
  intros; set (An := fun N:nat => / INR (fact N) * x ^ N).
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
0 < exp x
  cut (Un_cv (fun n:nat => sum_f_R0 An n) (exp x)).
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x) ->
0 < exp x
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
  intro; apply Rlt_le_trans with (sum_f_R0 An 0).
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
0 < sum_f_R0 An 0
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
sum_f_R0 An 0 <= exp x
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
  unfold An; simpl; rewrite Rinv_1; rewrite Rmult_1_r;
    apply Rlt_0_1.
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
sum_f_R0 An 0 <= exp x
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
  apply sum_incr.
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
forall n : nat, 0 <= An n
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
  assumption.
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
forall n : nat, 0 <= An n
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
  intro; unfold An; left; apply Rmult_lt_0_compat.
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) (exp x)
n:nat
0 < / INR (fact n)
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) (exp x)
n:nat
0 < x ^ n
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
  apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
H0:Un_cv (fun n0 : nat => sum_f_R0 An n0) (exp x)
n:nat
0 < x ^ n
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
  apply (pow_lt _ n H).
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
Un_cv (fun n : nat => sum_f_R0 An n) (exp x)
  unfold exp; unfold projT1; case (exist_exp x); intro.
x:R
H:0 < x
An:=fun N : nat => / INR (fact N) * x ^ N:nat -> R
x0:R
forall e : exp_in x x0,
Un_cv (fun n : nat => sum_f_R0 An n)
  (proj1_sig (exist (fun l : R => exp_in x l) x0 e))
  unfold exp_in; unfold infinite_sum, Un_cv; trivial.
Qed.

(**********)
Lemma exp_pos : forall x:R, 0 < exp x.
forall x : R, 0 < exp x
Proof.
forall x : R, 0 < exp x
  intro; destruct (total_order_T 0 x) as [[Hlt|<-]|Hgt].
x:R
Hlt:0 < x
0 < exp x
0 < exp 0
x:R
Hgt:0 > x
0 < exp x
  apply (exp_pos_pos _ Hlt).
0 < exp 0
x:R
Hgt:0 > x
0 < exp x
  rewrite exp_0; apply Rlt_0_1.
x:R
Hgt:0 > x
0 < exp x
  replace (exp x) with (1 / exp (- x)).
x:R
Hgt:0 > x
0 < 1 / exp (- x)
x:R
Hgt:0 > x
1 / exp (- x) = exp x
  unfold Rdiv; apply Rmult_lt_0_compat.
x:R
Hgt:0 > x
0 < 1
x:R
Hgt:0 > x
0 < / exp (- x)
x:R
Hgt:0 > x
1 / exp (- x) = exp x
  apply Rlt_0_1.
x:R
Hgt:0 > x
0 < / exp (- x)
x:R
Hgt:0 > x
1 / exp (- x) = exp x
  apply Rinv_0_lt_compat; apply exp_pos_pos.
x:R
Hgt:0 > x
0 < - x
x:R
Hgt:0 > x
1 / exp (- x) = exp x
  apply (Ropp_0_gt_lt_contravar _ Hgt).
x:R
Hgt:0 > x
1 / exp (- x) = exp x
  cut (exp (- x) <> 0).
x:R
Hgt:0 > x
exp (- x) <> 0 -> 1 / exp (- x) = exp x
x:R
Hgt:0 > x
exp (- x) <> 0
  intro; unfold Rdiv; apply Rmult_eq_reg_l with (exp (- x)).
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) * (1 * / exp (- x)) = exp (- x) * exp x
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) <> 0
x:R
Hgt:0 > x
exp (- x) <> 0
  rewrite Rmult_1_l; rewrite <- Rinv_r_sym.
x:R
Hgt:0 > x
H:exp (- x) <> 0
1 = exp (- x) * exp x
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) <> 0
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) <> 0
x:R
Hgt:0 > x
exp (- x) <> 0
  rewrite <- exp_plus.
x:R
Hgt:0 > x
H:exp (- x) <> 0
1 = exp (- x + x)
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) <> 0
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) <> 0
x:R
Hgt:0 > x
exp (- x) <> 0
  rewrite Rplus_opp_l; rewrite exp_0; reflexivity.
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) <> 0
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) <> 0
x:R
Hgt:0 > x
exp (- x) <> 0
  apply H.
x:R
Hgt:0 > x
H:exp (- x) <> 0
exp (- x) <> 0
x:R
Hgt:0 > x
exp (- x) <> 0
  apply H.
x:R
Hgt:0 > x
exp (- x) <> 0
  assert (H := exp_plus x (- x)).
x:R
Hgt:0 > x
H:exp (x + - x) = exp x * exp (- x)
exp (- x) <> 0
  rewrite Rplus_opp_r in H; rewrite exp_0 in H.
x:R
Hgt:0 > x
H:1 = exp x * exp (- x)
exp (- x) <> 0
  red; intro; rewrite H0 in H.
x:R
Hgt:0 > x
H:1 = exp x * 0
H0:exp (- x) = 0
False
  rewrite Rmult_0_r in H.
x:R
Hgt:0 > x
H:1 = 0
H0:exp (- x) = 0
False
  elim R1_neq_R0; assumption.
Qed.

(* ((exp h)-1)/h -> 0 quand h->0 *)
Lemma derivable_pt_lim_exp_0 : derivable_pt_lim exp 0 1.
derivable_pt_lim exp 0 1
Proof.
derivable_pt_lim exp 0 1
  unfold derivable_pt_lim; intros.
eps:R
H:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
  set (fn := fun (N:nat) (x:R) => x ^ N / INR (fact (S N))).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
  cut (CVN_R fn).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
(forall x : R,
 {l : R | Un_cv (fun N : nat => SP fn N x) l}) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  intro cv; cut (forall n:nat, continuity (fn n)).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
(forall n : nat, continuity (fn n)) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  intro; cut (continuity (SFL fn cv)).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv) ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  intro; unfold continuity in H1.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  assert (H2 := H1 0).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:continuity_pt (SFL fn cv) 0
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold continuity_pt in H2; unfold continue_in in H2; unfold limit1_in in H2;
    unfold limit_in in H2; simpl in H2; unfold R_dist in H2.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  elim (H2 _ H); intros alp H3.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  elim H3; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  exists (mkposreal _ H4); intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs ((exp (0 + h) - exp 0) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite Rplus_0_l; rewrite exp_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs ((exp h - 1) / h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace ((exp h - 1) / h) with (SFL fn cv h).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (SFL fn cv h - 1) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace 1 with (SFL fn cv 0).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (SFL fn cv h - SFL fn cv 0) < eps
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv 0 = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply H5.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
D_x no_cond 0 h /\ Rabs (h - 0) < alp
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv 0 = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  split.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
D_x no_cond 0 h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (h - 0) < alp
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv 0 = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold D_x, no_cond; split.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
True
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
0 <> h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (h - 0) < alp
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv 0 = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  trivial.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
0 <> h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (h - 0) < alp
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv 0 = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply (not_eq_sym H6).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
Rabs (h - 0) < alp
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv 0 = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite Rminus_0_r; apply H7.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv 0 = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold SFL.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
(let (a, _) := cv 0 in a) = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  case (cv 0) as (x,Hu).
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n : nat, continuity (fn n)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
x = 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  eapply UL_sequence.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n : nat, continuity (fn n)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
Un_cv ?Un x
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n : nat, continuity (fn n)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
Un_cv ?Un 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply Hu.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n : nat, continuity (fn n)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
Un_cv (fun N : nat => SP fn N 0) 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold Un_cv, SP in |- *.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n : nat, continuity (fn n)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (sum_f_R0 (fun k : nat => fn k 0) n) 1 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  intros; exists 1%nat; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
R_dist (sum_f_R0 (fun k : nat => fn k 0) n) 1 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold R_dist; rewrite decomp_sum.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
Rabs
  (fn 0%nat 0 +
   sum_f_R0 (fun i : nat => fn (S i) 0)
     (Init.Nat.pred n) - 1) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite (Rplus_comm (fn 0%nat 0)).
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
Rabs
  (sum_f_R0 (fun i : nat => fn (S i) 0)
     (Init.Nat.pred n) + 
   fn 0%nat 0 - 1) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace (fn 0%nat 0) with 1.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
Rabs
  (sum_f_R0 (fun i : nat => fn (S i) 0)
     (Init.Nat.pred n) + 1 - 1) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = fn 0%nat 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r;
    rewrite Rplus_0_r.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
Rabs
  (sum_f_R0 (fun i : nat => fn (S i) 0)
     (Init.Nat.pred n)) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = fn 0%nat 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace (sum_f_R0 (fun i:nat => fn (S i) 0) (pred n)) with 0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
Rabs 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
0 =
sum_f_R0 (fun i : nat => fn (S i) 0) (Init.Nat.pred n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = fn 0%nat 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite Rabs_R0; apply H8.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
0 =
sum_f_R0 (fun i : nat => fn (S i) 0) (Init.Nat.pred n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = fn 0%nat 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  symmetry ; apply sum_eq_R0; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n1 : nat, continuity (fn n1)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
n0:nat
H10:(n0 <= Init.Nat.pred n)%nat
fn (S n0) 0 = 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = fn 0%nat 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold fn.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n1 : nat, continuity (fn n1)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
n0:nat
H10:(n0 <= Init.Nat.pred n)%nat
0 ^ S n0 / INR (fact (S (S n0))) = 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = fn 0%nat 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  simpl.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n1 : nat, continuity (fn n1)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
n0:nat
H10:(n0 <= Init.Nat.pred n)%nat
0 * 0 ^ n0 /
INR
  (fact n0 + n0 * fact n0 +
   (fact n0 + n0 * fact n0 +
    n0 * (fact n0 + n0 * fact n0))) = 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = fn 0%nat 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold Rdiv; do 2 rewrite Rmult_0_l; reflexivity.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = fn 0%nat 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold fn; simpl.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
1 = 1 / 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold Rdiv; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x0 : R,
{l : R | Un_cv (fun N : nat => SP fn N x0) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x0 : R, continuity_pt (SFL fn cv) x0
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp0 ->
   Rabs (SFL fn cv x0 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
 Rabs (SFL fn cv x0 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
Rabs (SFL fn cv x0 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x:R
Hu:Un_cv (fun N : nat => SP fn N 0) x
eps0:R
H8:eps0 > 0
n:nat
H9:(n >= 1)%nat
(0 < n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H9 ].
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
SFL fn cv h = (exp h - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold SFL, exp.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
H1:forall x : R, continuity_pt (SFL fn cv) x
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond 0 x /\ Rabs (x - 0) < alp0 ->
   Rabs (SFL fn cv x - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x : R,
 D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
 Rabs (SFL fn cv x - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x : R,
D_x no_cond 0 x /\ Rabs (x - 0) < alp ->
Rabs (SFL fn cv x - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
(let (a, _) := cv h in a) =
(proj1_sig (exist_exp h) - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  case (cv h) as (x0,Hu); case (exist_exp h) as (x,Hexp); simpl.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:exp_in h x
x0 = (x - 1) / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  eapply UL_sequence.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:exp_in h x
Un_cv ?Un x0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:exp_in h x
Un_cv ?Un ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply Hu.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps0)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:exp_in h x
Un_cv (fun N : nat => SP fn N h) ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold Un_cv; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:exp_in h x
eps0:R
H8:eps0 > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (SP fn n h) ((x - 1) / h) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold exp_in, infinite_sum in Hexp.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (SP fn n h) ((x - 1) / h) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  cut (0 < eps0 * Rabs h).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (SP fn n h) ((x - 1) / h) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  intro; elim (Hexp _ H9); intros N0 H10.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n : nat,
(n >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n)
  x < eps0 * Rabs h
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (SP fn n h) ((x - 1) / h) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  exists N0; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
R_dist (SP fn n h) ((x - 1) / h) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold R_dist.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs (SP fn n h - (x - 1) / h) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply Rmult_lt_reg_l with (Rabs h).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
0 < Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs h * Rabs (SP fn n h - (x - 1) / h) <
Rabs h * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply Rabs_pos_lt; assumption.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs h * Rabs (SP fn n h - (x - 1) / h) <
Rabs h * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite <- Rabs_mult.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs (h * (SP fn n h - (x - 1) / h)) < Rabs h * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite Rmult_minus_distr_l.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs (h * SP fn n h - h * ((x - 1) / h)) <
Rabs h * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace (h * ((x - 1) / h)) with (x - 1).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs (h * SP fn n h - (x - 1)) < Rabs h * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold R_dist in H10.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs (h * SP fn n h - (x - 1)) < Rabs h * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace (h * SP fn n h - (x - 1)) with
  (sum_f_R0 (fun i:nat => / INR (fact i) * h ^ i) (S n) - x).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs
  (sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i)
     (S n) - x) < Rabs h * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i) (S n) -
x = h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite (Rmult_comm (Rabs h)).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
Rabs
  (sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i)
     (S n) - x) < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i) (S n) -
x = h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply H10.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i) (S n) -
x = h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold ge.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(N0 <= S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i) (S n) -
x = h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply le_trans with (S N0).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(N0 <= S N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(S N0 <= S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i) (S n) -
x = h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply le_n_Sn.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(S N0 <= S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i) (S n) -
x = h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply le_n_S; apply H11.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact i) * h ^ i) (S n) -
x = h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite decomp_sum.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
/ INR (fact 0) * h ^ 0 +
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  (Init.Nat.pred (S n)) - x = 
h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace (/ INR (fact 0) * h ^ 0) with 1.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 +
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  (Init.Nat.pred (S n)) - x = 
h * SP fn n h - (x - 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold Rminus.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 +
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  (Init.Nat.pred (S n)) + 
- x = h * SP fn n h + - (x + - (1))
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite Ropp_plus_distr.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 +
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  (Init.Nat.pred (S n)) + 
- x = h * SP fn n h + (- x + - - (1))
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite Ropp_involutive.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 +
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  (Init.Nat.pred (S n)) + 
- x = h * SP fn n h + (- x + 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite <- (Rplus_comm (- x)).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
- x +
(1 +
 sum_f_R0
   (fun i : nat => / INR (fact (S i)) * h ^ S i)
   (Init.Nat.pred (S n))) = 
h * SP fn n h + (- x + 1)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite <- (Rplus_comm (- x + 1)).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
- x +
(1 +
 sum_f_R0
   (fun i : nat => / INR (fact (S i)) * h ^ S i)
   (Init.Nat.pred (S n))) = 
- x + 1 + h * SP fn n h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite Rplus_assoc; repeat apply Rplus_eq_compat_l.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  (Init.Nat.pred (S n)) = 
h * SP fn n h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace (pred (S n)) with n; [ idtac | reflexivity ].
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  n = h * SP fn n h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold SP.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  n = h * sum_f_R0 (fun k : nat => fn k h) n
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite scal_sum.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
sum_f_R0 (fun i : nat => / INR (fact (S i)) * h ^ S i)
  n = sum_f_R0 (fun i : nat => fn i h * h) n
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply sum_eq; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i0 : nat =>
        / INR (fact i0) * h ^ i0) n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i0 : nat => / INR (fact i0) * h ^ i0)
     n0 - x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
i:nat
H12:(i <= n)%nat
/ INR (fact (S i)) * h ^ S i = fn i h * h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold fn.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i0 : nat =>
        / INR (fact i0) * h ^ i0) n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i0 : nat => / INR (fact i0) * h ^ i0)
     n0 - x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
i:nat
H12:(i <= n)%nat
/ INR (fact (S i)) * h ^ S i =
h ^ i / INR (fact (S i)) * h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace (h ^ S i) with (h * h ^ i).
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i0 : nat =>
        / INR (fact i0) * h ^ i0) n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i0 : nat => / INR (fact i0) * h ^ i0)
     n0 - x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
i:nat
H12:(i <= n)%nat
/ INR (fact (S i)) * (h * h ^ i) =
h ^ i / INR (fact (S i)) * h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i0 : nat =>
        / INR (fact i0) * h ^ i0) n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i0 : nat => / INR (fact i0) * h ^ i0)
     n0 - x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
i:nat
H12:(i <= n)%nat
h * h ^ i = h ^ S i
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold Rdiv; ring.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i0 : nat =>
        / INR (fact i0) * h ^ i0) n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i0 : nat => / INR (fact i0) * h ^ i0)
     n0 - x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
i:nat
H12:(i <= n)%nat
h * h ^ i = h ^ S i
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  simpl; ring.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
1 = / INR (fact 0) * h ^ 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  simpl; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0 -
   x) < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
(0 < S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply lt_O_Sn.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  unfold Rdiv.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * ((x - 1) * / h)
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  rewrite <- Rmult_assoc.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
x - 1 = h * (x - 1) * / h
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  symmetry ; apply Rinv_r_simpl_m.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n0 : nat, continuity (fn n0)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n0) x < eps1
eps0:R
H8:eps0 > 0
H9:0 < eps0 * Rabs h
N0:nat
H10:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (sum_f_R0
     (fun i : nat => / INR (fact i) * h ^ i) n0)
  x < eps0 * Rabs h
n:nat
H11:(n >= N0)%nat
h <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  assumption.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0 * Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply Rmult_lt_0_compat.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply H8.
eps:R
H:0 < eps
fn:=fun (N : nat) (x1 : R) =>
x1 ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x1 : R,
{l : R | Un_cv (fun N : nat => SP fn N x1) l}
H0:forall n : nat, continuity (fn n)
H1:forall x1 : R, continuity_pt (SFL fn cv) x1
H2:forall eps1 : R,
eps1 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x1 : R,
   D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp0 ->
   Rabs (SFL fn cv x1 - SFL fn cv 0) < eps1)
alp:R
H3:alp > 0 /\
(forall x1 : R,
 D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
 Rabs (SFL fn cv x1 - SFL fn cv 0) < eps)
H4:alp > 0
H5:forall x1 : R,
D_x no_cond 0 x1 /\ Rabs (x1 - 0) < alp ->
Rabs (SFL fn cv x1 - SFL fn cv 0) < eps
h:R
H6:h <> 0
H7:Rabs h < {| pos := alp; cond_pos := H4 |}
x0:R
Hu:Un_cv (fun N : nat => SP fn N h) x0
x:R
Hexp:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (sum_f_R0
       (fun i : nat => / INR (fact i) * h ^ i)
       n) x < eps1
eps0:R
H8:eps0 > 0
0 < Rabs h
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply Rabs_pos_lt; assumption.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
H0:forall n : nat, continuity (fn n)
continuity (SFL fn cv)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply SFL_continuity; assumption.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
forall n : nat, continuity (fn n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  intro; unfold fn.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity (fun x : R => x ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  replace (fun x:R => x ^ n / INR (fact (S n))) with
  (pow_fct n / fct_cte (INR (fact (S n))))%F; [ idtac | reflexivity ].
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity (pow_fct n / fct_cte (INR (fact (S n))))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply continuity_div.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity (pow_fct n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity (fct_cte (INR (fact (S n))))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
forall x : R, fct_cte (INR (fact (S n))) x <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply derivable_continuous; apply (derivable_pow n).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
continuity (fct_cte (INR (fact (S n))))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
forall x : R, fct_cte (INR (fact (S n))) x <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply derivable_continuous; apply derivable_const.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
cv:forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
n:nat
forall x : R, fct_cte (INR (fact (S n))) x <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  intro; unfold fct_cte; apply INR_fact_neq_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
X:CVN_R fn
forall x : R,
{l : R | Un_cv (fun N : nat => SP fn N x) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  apply (CVN_R_CVS _ X).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
CVN_R fn
  assert (H0 := Alembert_exp).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
CVN_R fn
  unfold CVN_R.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
forall r : posreal, CVN_r fn r
  intro; unfold CVN_r.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{An : nat -> R &
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0 (fun k : nat => Rabs (An k)) n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y -> Rabs (fn n y) <= An n)}}
  exists (fun N:nat => r ^ N / INR (fact (S N))).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <= r ^ n / INR (fact (S n)))}
  cut
    { l:R |
        Un_cv
        (fun n:nat =>
          sum_f_R0 (fun k:nat => Rabs (r ^ k / INR (fact (S k)))) n) l }.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l} ->
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <= r ^ n / INR (fact (S n)))}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  intros (x,p).
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n) x
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <= r ^ n / INR (fact (S n)))}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  exists x; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n) x
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) x /\
(forall (n : nat) (y : R),
 Boule 0 r y ->
 Rabs (fn n y) <= r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  split.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n) x
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) x
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n) x
forall (n : nat) (y : R),
Boule 0 r y ->
Rabs (fn n y) <= r ^ n / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  apply p.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
x:R
p:Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n) x
forall (n : nat) (y : R),
Boule 0 r y ->
Rabs (fn n y) <= r ^ n / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  unfold Boule; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs (y - 0) < r
Rabs (fn n y) <= r ^ n / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  rewrite Rminus_0_r in H1.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
Rabs (fn n y) <= r ^ n / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  unfold fn.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
Rabs (y ^ n / INR (fact (S n))) <=
r ^ n / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  unfold Rdiv; rewrite Rabs_mult.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
Rabs (y ^ n) * Rabs (/ INR (fact (S n))) <=
r ^ n * / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  cut (0 < INR (fact (S n))).
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n)) ->
Rabs (y ^ n) * Rabs (/ INR (fact (S n))) <=
r ^ n * / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  intro.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
Rabs (y ^ n) * Rabs (/ INR (fact (S n))) <=
r ^ n * / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  rewrite (Rabs_right (/ INR (fact (S n)))).
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
Rabs (y ^ n) * / INR (fact (S n)) <=
r ^ n * / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
/ INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  do 2 rewrite <- (Rmult_comm (/ INR (fact (S n)))).
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
/ INR (fact (S n)) * Rabs (y ^ n) <=
/ INR (fact (S n)) * r ^ n
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
/ INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  apply Rmult_le_compat_l.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
0 <= / INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
Rabs (y ^ n) <= r ^ n
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
/ INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  left; apply Rinv_0_lt_compat; apply H2.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
Rabs (y ^ n) <= r ^ n
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
/ INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  rewrite <- RPow_abs.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
Rabs y ^ n <= r ^ n
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
/ INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  apply pow_maj_Rabs.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
Rabs (Rabs y) <= r
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
/ INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  rewrite Rabs_Rabsolu; left; apply H1.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
H2:0 < INR (fact (S n))
/ INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  apply Rle_ge; left; apply Rinv_0_lt_compat; apply H2.
eps:R
H:0 < eps
fn:=fun (N : nat) (x0 : R) =>
x0 ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
x:R
p:Un_cv
  (fun n0 : nat =>
   sum_f_R0
     (fun k : nat =>
      Rabs (r ^ k / INR (fact (S k)))) n0) x
n:nat
y:R
H1:Rabs y < r
0 < INR (fact (S n))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  apply INR_fact_lt_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
  cut ((r:R) <> 0).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0 ->
{l : R |
Un_cv
  (fun n : nat =>
   sum_f_R0
     (fun k : nat => Rabs (r ^ k / INR (fact (S k))))
     n) l}
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  intro; apply Alembert_C2.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
H1:(r : R) <> 0
forall n : nat, Rabs (r ^ n / INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
H1:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  intro; apply Rabs_no_R0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
H1:(r : R) <> 0
n:nat
r ^ n / INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
H1:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  unfold Rdiv; apply prod_neq_R0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
H1:(r : R) <> 0
n:nat
r ^ n <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
H1:(r : R) <> 0
n:nat
/ INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
H1:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply pow_nonzero; assumption.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n0 : nat =>
   Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0
r:posreal
H1:(r : R) <> 0
n:nat
/ INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
H1:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rinv_neq_0_compat; apply INR_fact_neq_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
H1:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  unfold Un_cv in H0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs (/ INR (fact (S n)) * / / INR (fact n)))
    0 < eps0
r:posreal
H1:(r : R) <> 0
Un_cv
  (fun n : nat =>
   Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  unfold Un_cv; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs (/ INR (fact (S n)) * / / INR (fact n)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs (r ^ S n / INR (fact (S (S n)))) /
        Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  cut (0 < eps0 / r);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
        [ assumption | apply Rinv_0_lt_compat; apply (cond_pos r) ] ].
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs (/ INR (fact (S n)) * / / INR (fact n)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs (r ^ S n / INR (fact (S (S n)))) /
        Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  elim (H0 _ H3); intros N0 H4.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs (/ INR (fact (S n)) * / / INR (fact n)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n : nat,
(n >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0 < eps0 / r
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist
    (Rabs
       (Rabs (r ^ S n / INR (fact (S (S n)))) /
        Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  exists N0; intros.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
R_dist
  (Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  cut (S n >= N0)%nat.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat ->
R_dist
  (Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  intro hyp_sn.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
R_dist
  (Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  assert (H6 := H4 _ hyp_sn).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:R_dist
  (Rabs
     (/ INR (fact (S (S n))) *
      / / INR (fact (S n)))) 0 < 
eps0 / r
R_dist
  (Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  unfold R_dist in H6; rewrite Rminus_0_r in H6.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (Rabs
     (/ INR (fact (S (S n))) *
      / / INR (fact (S n)))) < 
eps0 / r
R_dist
  (Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite Rabs_Rabsolu in H6.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
R_dist
  (Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) 0 < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  unfold R_dist; rewrite Rminus_0_r.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs
  (Rabs
     (Rabs (r ^ S n / INR (fact (S (S n)))) /
      Rabs (r ^ n / INR (fact (S n))))) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite Rabs_Rabsolu.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs
  (Rabs (r ^ S n / INR (fact (S (S n)))) /
   Rabs (r ^ n / INR (fact (S n)))) < eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  replace
  (Rabs (r ^ S n / INR (fact (S (S n)))) / Rabs (r ^ n / INR (fact (S n))))
    with (r * / INR (fact (S (S n))) * / / INR (fact (S n))).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs
  (r * / INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite Rmult_assoc; rewrite Rabs_mult.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs r *
Rabs (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite (Rabs_right r).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r *
Rabs (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rmult_lt_reg_l with (/ r).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
0 < / r
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ r *
(r *
 Rabs (/ INR (fact (S (S n))) * / / INR (fact (S n)))) <
/ r * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rinv_0_lt_compat; apply (cond_pos r).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ r *
(r *
 Rabs (/ INR (fact (S (S n))) * / / INR (fact (S n)))) <
/ r * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
1 *
Rabs (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
/ r * eps0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite Rmult_1_l; rewrite <- (Rmult_comm eps0).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 * / r
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply H6.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  assumption.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rle_ge; left; apply (cond_pos r).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n / INR (fact (S (S n)))) /
Rabs (r ^ n / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  unfold Rdiv.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n * / INR (fact (S (S n)))) *
/ Rabs (r ^ n * / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  repeat rewrite Rabs_mult.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n) * Rabs (/ INR (fact (S (S n)))) *
/ (Rabs (r ^ n) * Rabs (/ INR (fact (S n))))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  repeat rewrite Rabs_Rinv.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n) * / Rabs (INR (fact (S (S n)))) *
/ (Rabs (r ^ n) * / Rabs (INR (fact (S n))))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite Rinv_mult_distr.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
Rabs (r ^ S n) * / Rabs (INR (fact (S (S n)))) *
(/ Rabs (r ^ n) * / / Rabs (INR (fact (S n))))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  repeat rewrite Rabs_right.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * / / INR (fact (S n)) =
r ^ S n * / INR (fact (S (S n))) *
(/ r ^ n * / / INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite Rinv_involutive.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * / INR (fact (S (S n))) * INR (fact (S n)) =
r ^ S n * / INR (fact (S (S n))) *
(/ r ^ n * INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite (Rmult_comm r).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ INR (fact (S (S n))) * r * INR (fact (S n)) =
r ^ S n * / INR (fact (S (S n))) *
(/ r ^ n * INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite (Rmult_comm (r ^ S n)).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ INR (fact (S (S n))) * r * INR (fact (S n)) =
/ INR (fact (S (S n))) * r ^ S n *
(/ r ^ n * INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  repeat rewrite Rmult_assoc.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ INR (fact (S (S n))) * (r * INR (fact (S n))) =
/ INR (fact (S (S n))) *
(r ^ S n * (/ r ^ n * INR (fact (S n))))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rmult_eq_compat_l.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r * INR (fact (S n)) =
r ^ S n * (/ r ^ n * INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite (Rmult_comm r).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) * r =
r ^ S n * (/ r ^ n * INR (fact (S n)))
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite <- Rmult_assoc; rewrite <- (Rmult_comm (INR (fact (S n)))).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) * r =
INR (fact (S n)) * (r ^ S n * / r ^ n)
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rmult_eq_compat_l.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r = r ^ S n * / r ^ n
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  simpl.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r = r * r ^ n * / r ^ n
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r = r * 1
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  ring.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply pow_nonzero; assumption.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply INR_fact_neq_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rle_ge; left; apply INR_fact_lt_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rle_ge; left; apply pow_lt; apply (cond_pos r).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rle_ge; left; apply INR_fact_lt_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
r ^ S n >= 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rle_ge; left; apply pow_lt; apply (cond_pos r).
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
Rabs (r ^ n) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rabs_no_R0; apply pow_nonzero; assumption.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
/ Rabs (INR (fact (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply Rinv_neq_0_compat; apply Rabs_no_R0; apply INR_fact_neq_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S n)) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply INR_fact_neq_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
hyp_sn:(S n >= N0)%nat
H6:Rabs
  (/ INR (fact (S (S n))) * / / INR (fact (S n))) <
eps0 / r
INR (fact (S (S n))) <> 0
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply INR_fact_neq_0.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(S n >= N0)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  unfold ge; apply le_trans with n.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(N0 <= n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(n <= S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply H5.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:forall eps1 : R,
eps1 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist
    (Rabs
       (/ INR (fact (S n0)) * / / INR (fact n0)))
    0 < eps1
r:posreal
H1:(r : R) <> 0
eps0:R
H2:eps0 > 0
H3:0 < eps0 / r
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
R_dist
  (Rabs (/ INR (fact (S n0)) * / / INR (fact n0)))
  0 < eps0 / r
n:nat
H5:(n >= N0)%nat
(n <= S n)%nat
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  apply le_n_Sn.
eps:R
H:0 < eps
fn:=fun (N : nat) (x : R) => x ^ N / INR (fact (S N)):nat -> R -> R
H0:Un_cv
  (fun n : nat =>
   Rabs (/ INR (fact (S n)) * / / INR (fact n)))
  0
r:posreal
(r : R) <> 0
  assert (H1 := cond_pos r); red; intro; rewrite H2 in H1;
    elim (Rlt_irrefl _ H1).
Qed.

(**********)
Lemma derivable_pt_lim_exp : forall x:R, derivable_pt_lim exp x (exp x).
forall x : R, derivable_pt_lim exp x (exp x)
Proof.
forall x : R, derivable_pt_lim exp x (exp x)
  intro; assert (H0 := derivable_pt_lim_exp_0).
x:R
H0:derivable_pt_lim exp 0 1
derivable_pt_lim exp x (exp x)
  unfold derivable_pt_lim in H0; unfold derivable_pt_lim; intros.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps0
eps:R
H:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (x + h) - exp x) / h - exp x) < eps
  cut (0 < eps / exp x);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
        [ apply H | apply Rinv_0_lt_compat; apply exp_pos ] ].
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (x + h) - exp x) / h - exp x) < eps
  elim (H0 _ H1); intros del H2.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (0 + h) - exp 0) / h - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h : R,
h <> 0 ->
Rabs h < del ->
Rabs ((exp (0 + h) - exp 0) / h - 1) <
eps / exp x
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((exp (x + h) - exp x) / h - exp x) < eps
  exists del; intros.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
Rabs ((exp (x + h) - exp x) / h - exp x) < eps
  assert (H5 := H2 _ H3 H4).
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp (0 + h) - exp 0) / h - 1) <
eps / exp x
Rabs ((exp (x + h) - exp x) / h - exp x) < eps
  rewrite Rplus_0_l in H5; rewrite exp_0 in H5.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
Rabs ((exp (x + h) - exp x) / h - exp x) < eps
  replace ((exp (x + h) - exp x) / h - exp x) with
  (exp x * ((exp h - 1) / h - 1)).
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
Rabs (exp x * ((exp h - 1) / h - 1)) < eps
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  rewrite Rabs_mult; rewrite (Rabs_right (exp x)).
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * Rabs ((exp h - 1) / h - 1) < eps
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x >= 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  apply Rmult_lt_reg_l with (/ exp x).
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
0 < / exp x
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
/ exp x * (exp x * Rabs ((exp h - 1) / h - 1)) <
/ exp x * eps
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x >= 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  apply Rinv_0_lt_compat; apply exp_pos.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
/ exp x * (exp x * Rabs ((exp h - 1) / h - 1)) <
/ exp x * eps
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x >= 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
1 * Rabs ((exp h - 1) / h - 1) < / exp x * eps
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x <> 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x >= 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  rewrite Rmult_1_l; rewrite <- (Rmult_comm eps).
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
Rabs ((exp h - 1) / h - 1) < eps * / exp x
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x <> 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x >= 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  apply H5.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x <> 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x >= 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  assert (H6 := exp_pos x); red; intro; rewrite H7 in H6;
    elim (Rlt_irrefl _ H6).
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x >= 0
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  apply Rle_ge; left; apply exp_pos.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h - 1) =
(exp (x + h) - exp x) / h - exp x
  rewrite Rmult_minus_distr_l.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
exp x * ((exp h - 1) / h) - exp x * 1 =
(exp (x + h) - exp x) / h - exp x
  rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rmult_assoc;
    rewrite Rmult_minus_distr_l.
x:R
H0:forall eps0 : R,
0 < eps0 ->
exists delta : posreal,
  forall h0 : R,
  h0 <> 0 ->
  Rabs h0 < delta ->
  Rabs ((exp (0 + h0) - exp 0) / h0 - 1) < eps0
eps:R
H:0 < eps
H1:0 < eps / exp x
del:posreal
H2:forall h0 : R,
h0 <> 0 ->
Rabs h0 < del ->
Rabs ((exp (0 + h0) - exp 0) / h0 - 1) <
eps / exp x
h:R
H3:h <> 0
H4:Rabs h < del
H5:Rabs ((exp h - 1) / h - 1) < eps / exp x
(exp x * exp h - exp x * 1) * / h - exp x =
(exp (x + h) - exp x) * / h - exp x
  rewrite Rmult_1_r; rewrite exp_plus; reflexivity.
Qed.