(* -*- coding: utf-8 -*- *)
(************************************************************************)
(*         *   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 Rfunctions.
Require Import SeqSeries.
Require Import Ranalysis_reg.
Require Import Rbase.
Require Import RiemannInt_SF.
Require Import Max.
Local Open Scope R_scope.

Set Implicit Arguments.

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

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

Definition Riemann_integrable (f:R -> R) (a b:R) : Type :=
  forall eps:posreal,
    { phi:StepFun a b &
      { psi:StepFun a b |
        (forall t:R,
          Rmin a b <= t <= Rmax a b -> Rabs (f t - phi t) <= psi t) /\
        Rabs (RiemannInt_SF psi) < eps } }.

Definition phi_sequence (un:nat -> posreal) (f:R -> R)
  (a b:R) (pr:Riemann_integrable f a b) (n:nat) :=
  projT1 (pr (un n)).

Lemma phi_sequence_prop :
  forall (un:nat -> posreal) (f:R -> R) (a b:R) (pr:Riemann_integrable f a b)
    (N:nat),
    { psi:StepFun a b |
      (forall t:R,
        Rmin a b <= t <= Rmax a b ->
        Rabs (f t - phi_sequence un pr N t) <= psi t) /\
      Rabs (RiemannInt_SF psi) < un N }.
forall (un : nat -> posreal) (f : R -> R) (a b : R)
  (pr : Riemann_integrable f a b) (N : nat),
{psi : StepFun a b |
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr N t) <= psi t) /\
Rabs (RiemannInt_SF psi) < un N}
Proof.
forall (un : nat -> posreal) (f : R -> R) (a b : R)
  (pr : Riemann_integrable f a b) (N : nat),
{psi : StepFun a b |
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr N t) <= psi t) /\
Rabs (RiemannInt_SF psi) < un N}
  intros; apply (projT2 (pr (un N))).
Qed.

Lemma RiemannInt_P1 :
  forall (f:R -> R) (a b:R),
    Riemann_integrable f a b -> Riemann_integrable f b a.
forall (f : R -> R) (a b : R),
Riemann_integrable f a b -> Riemann_integrable f b a
Proof.
forall (f : R -> R) (a b : R),
Riemann_integrable f a b -> Riemann_integrable f b a
  unfold Riemann_integrable; intros; elim (X eps); clear X; intros.
f:R -> R
a, b:R
eps:posreal
x:StepFun a b
p:{psi : StepFun a b |
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - x t) <= psi t) /\
Rabs (RiemannInt_SF psi) < eps}
{phi : StepFun b a &
{psi : StepFun b a |
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - phi t) <= psi t) /\
Rabs (RiemannInt_SF psi) < eps}}
  elim p; clear p; intros x0 p; exists (mkStepFun (StepFun_P6 (pre x)));
      exists (mkStepFun (StepFun_P6 (pre x0)));
        elim p; clear p; intros; split.
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
forall t : R,
Rmin b a <= t <= Rmax b a ->
Rabs
  (f t - {| fe := x; pre := StepFun_P6 (pre x) |} t) <=
{| fe := x0; pre := StepFun_P6 (pre x0) |} t
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
Rabs
  (RiemannInt_SF
     {| fe := x0; pre := StepFun_P6 (pre x0) |}) < eps
  intros; apply (H t); elim H1; clear H1; intros; split;
    [ apply Rle_trans with (Rmin b a); try assumption; right;
      unfold Rmin
      | apply Rle_trans with (Rmax b a); try assumption; right;
        unfold Rmax ];
    (case (Rle_dec a b); case (Rle_dec b a); intros;
      try reflexivity || apply Rle_antisym;
        [ assumption | assumption | auto with real | auto with real ]).
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
Rabs
  (RiemannInt_SF
     {| fe := x0; pre := StepFun_P6 (pre x0) |}) < eps
  generalize H0; unfold RiemannInt_SF; case (Rle_dec a b);
    case (Rle_dec b a); intros;
      (replace
        (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre x0))))
          (subdivision (mkStepFun (StepFun_P6 (pre x0))))) with
        (Int_SF (subdivision_val x0) (subdivision x0));
        [ idtac
          | apply StepFun_P17 with (fe x0) a b;
            [ apply StepFun_P1
              | apply StepFun_P2;
                apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre x0)))) ] ]).
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
r:b <= a
r0:a <= b
H1:Rabs
  (Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (Int_SF (subdivision_val x0) (subdivision x0)) <
eps
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
n:~ b <= a
r:a <= b
H1:Rabs
  (Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
r:b <= a
n:~ a <= b
H1:Rabs
  (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (Int_SF (subdivision_val x0) (subdivision x0)) <
eps
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
n:~ b <= a
n0:~ a <= b
H1:Rabs
  (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
  apply H1.
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
n:~ b <= a
r:a <= b
H1:Rabs
  (Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
r:b <= a
n:~ a <= b
H1:Rabs
  (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (Int_SF (subdivision_val x0) (subdivision x0)) <
eps
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
n:~ b <= a
n0:~ a <= b
H1:Rabs
  (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
  rewrite Rabs_Ropp; apply H1.
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
r:b <= a
n:~ a <= b
H1:Rabs
  (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (Int_SF (subdivision_val x0) (subdivision x0)) <
eps
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
n:~ b <= a
n0:~ a <= b
H1:Rabs
  (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
  rewrite Rabs_Ropp in H1; apply H1.
f:R -> R
a, b:R
eps:posreal
x, x0:StepFun a b
H:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - x t) <= x0 t
H0:Rabs (RiemannInt_SF x0) < eps
n:~ b <= a
n0:~ a <= b
H1:Rabs
  (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
Rabs (- Int_SF (subdivision_val x0) (subdivision x0)) <
eps
  apply H1.
Qed.

Lemma RiemannInt_P2 :
  forall (f:R -> R) (a b:R) (un:nat -> posreal) (vn wn:nat -> StepFun a b),
    Un_cv un 0 ->
    a <= b ->
    (forall n:nat,
      (forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\
      Rabs (RiemannInt_SF (wn n)) < un n) ->
    { l:R | Un_cv (fun N:nat => RiemannInt_SF (vn N)) l }.
forall (f : R -> R) (a b : R) (un : nat -> posreal)
  (vn wn : nat -> StepFun a b),
Un_cv (fun x : nat => un x) 0 ->
a <= b ->
(forall n : nat,
 (forall t : R,
  Rmin a b <= t <= Rmax a b ->
  Rabs (f t - vn n t) <= wn n t) /\
 Rabs (RiemannInt_SF (wn n)) < un n) ->
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
Proof.
forall (f : R -> R) (a b : R) (un : nat -> posreal)
  (vn wn : nat -> StepFun a b),
Un_cv (fun x : nat => un x) 0 ->
a <= b ->
(forall n : nat,
 (forall t : R,
  Rmin a b <= t <= Rmax a b ->
  Rabs (f t - vn n t) <= wn n t) /\
 Rabs (RiemannInt_SF (wn n)) < un n) ->
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  intros; apply R_complete; unfold Un_cv in H; unfold Cauchy_crit;
    intros; assert (H3 : 0 < eps / 2).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (un n) 0 < eps0
H0:a <= b
H1:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
eps:R
H2:eps > 0
0 < eps / 2
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (un n) 0 < eps0
H0:a <= b
H1:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
eps:R
H2:eps > 0
H3:0 < eps / 2
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (RiemannInt_SF (vn n)) (RiemannInt_SF (vn m)) <
  eps
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (un n) 0 < eps0
H0:a <= b
H1:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
eps:R
H2:eps > 0
H3:0 < eps / 2
exists N : nat,
  forall n m : nat,
  (n >= N)%nat ->
  (m >= N)%nat ->
  R_dist (RiemannInt_SF (vn n)) (RiemannInt_SF (vn m)) <
  eps
  elim (H _ H3); intros N0 H4; exists N0; intros; unfold R_dist;
    unfold R_dist in H4; elim (H1 n); elim (H1 m); intros;
      replace (RiemannInt_SF (vn n) - RiemannInt_SF (vn m)) with
      (RiemannInt_SF (vn n) + -1 * RiemannInt_SF (vn m));
      [ idtac | ring ]; rewrite <- StepFun_P30;
        apply Rle_lt_trans with
          (RiemannInt_SF
            (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (vn n) (vn m)))))).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R => vn n x + -1 * vn m x;
     pre := StepFun_P28 (-1) (vn n) (vn m) |}) <=
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := fun x0 : R => vn n x0 + -1 * vn m x0;
           pre := StepFun_P28 (-1) (vn n) (vn m) |} x);
  pre := StepFun_P32
           {|
           fe := fun x : R => vn n x + -1 * vn m x;
           pre := StepFun_P28 (-1) (vn n) (vn m) |} |}
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := fun x0 : R => vn n x0 + -1 * vn m x0;
           pre := StepFun_P28 (-1) (vn n) (vn m) |} x);
  pre := StepFun_P32
           {|
           fe := fun x : R => vn n x + -1 * vn m x;
           pre := StepFun_P28 (-1) (vn n) (vn m) |} |} <
eps
  apply StepFun_P34; assumption.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := fun x0 : R => vn n x0 + -1 * vn m x0;
           pre := StepFun_P28 (-1) (vn n) (vn m) |} x);
  pre := StepFun_P32
           {|
           fe := fun x : R => vn n x + -1 * vn m x;
           pre := StepFun_P28 (-1) (vn n) (vn m) |} |} <
eps
  apply Rle_lt_trans with
    (RiemannInt_SF (mkStepFun (StepFun_P28 1 (wn n) (wn m)))).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := fun x0 : R => vn n x0 + -1 * vn m x0;
           pre := StepFun_P28 (-1) (vn n) (vn m) |} x);
  pre := StepFun_P32
           {|
           fe := fun x : R => vn n x + -1 * vn m x;
           pre := StepFun_P28 (-1) (vn n) (vn m) |} |} <=
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |}
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  apply StepFun_P37; try assumption.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
forall x : R,
a < x < b ->
{|
fe := fun x0 : R =>
      Rabs
        ({|
         fe := fun x1 : R => vn n x1 + -1 * vn m x1;
         pre := StepFun_P28 (-1) (vn n) (vn m) |} x0);
pre := StepFun_P32
         {|
         fe := fun x0 : R => vn n x0 + -1 * vn m x0;
         pre := StepFun_P28 (-1) (vn n) (vn m) |} |} x <=
{|
fe := fun x0 : R => wn n x0 + 1 * wn m x0;
pre := StepFun_P28 1 (wn n) (wn m) |} x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  intros; simpl;
    apply Rle_trans with (Rabs (vn n x - f x) + Rabs (f x - vn m x)).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
Rabs (vn n x + -1 * vn m x) <=
Rabs (vn n x - f x) + Rabs (f x - vn m x)
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
Rabs (vn n x - f x) + Rabs (f x - vn m x) <=
wn n x + 1 * wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  replace (vn n x + -1 * vn m x) with (vn n x - f x + (f x - vn m x));
  [ apply Rabs_triang | ring ].
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
Rabs (vn n x - f x) + Rabs (f x - vn m x) <=
wn n x + 1 * wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  assert (H12 : Rmin a b = a).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
Rmin a b = a
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
H12:Rmin a b = a
Rabs (vn n x - f x) + Rabs (f x - vn m x) <=
wn n x + 1 * wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  unfold Rmin; decide (Rle_dec a b) with H0; reflexivity.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
H12:Rmin a b = a
Rabs (vn n x - f x) + Rabs (f x - vn m x) <=
wn n x + 1 * wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  assert (H13 : Rmax a b = b).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
H12:Rmin a b = a
Rmax a b = b
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
H12:Rmin a b = a
H13:Rmax a b = b
Rabs (vn n x - f x) + Rabs (f x - vn m x) <=
wn n x + 1 * wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  unfold Rmax; decide (Rle_dec a b) with H0; reflexivity.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:a < x < b
H12:Rmin a b = a
H13:Rmax a b = b
Rabs (vn n x - f x) + Rabs (f x - vn m x) <=
wn n x + 1 * wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  rewrite <- H12 in H11; rewrite <- H13 in H11 at 2;
    rewrite Rmult_1_l; apply Rplus_le_compat.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:Rmin a b < x < Rmax a b
H12:Rmin a b = a
H13:Rmax a b = b
Rabs (vn n x - f x) <= wn n x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:Rmin a b < x < Rmax a b
H12:Rmin a b = a
H13:Rmax a b = b
Rabs (f x - vn m x) <= wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:Rmin a b < x < Rmax a b
H12:Rmin a b = a
H13:Rmax a b = b
Rmin a b <= x <= Rmax a b
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:Rmin a b < x < Rmax a b
H12:Rmin a b = a
H13:Rmax a b = b
Rabs (f x - vn m x) <= wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  elim H11; intros; split; left; assumption.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:Rmin a b < x < Rmax a b
H12:Rmin a b = a
H13:Rmax a b = b
Rabs (f x - vn m x) <= wn m x
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  apply H7.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
x:R
H11:Rmin a b < x < Rmax a b
H12:Rmin a b = a
H13:Rmax a b = b
Rmin a b <= x <= Rmax a b
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  elim H11; intros; split; left; assumption.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF
  {|
  fe := fun x : R => wn n x + 1 * wn m x;
  pre := StepFun_P28 1 (wn n) (wn m) |} < eps
  rewrite StepFun_P30; rewrite Rmult_1_l; apply Rlt_trans with (un n + un m).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF (wn n) + RiemannInt_SF (wn m) <
un n + un m
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
un n + un m < eps
  apply Rle_lt_trans with
    (Rabs (RiemannInt_SF (wn n)) + Rabs (RiemannInt_SF (wn m))).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
RiemannInt_SF (wn n) + RiemannInt_SF (wn m) <=
Rabs (RiemannInt_SF (wn n)) +
Rabs (RiemannInt_SF (wn m))
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
Rabs (RiemannInt_SF (wn n)) +
Rabs (RiemannInt_SF (wn m)) < un n + un m
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
un n + un m < eps
  apply Rplus_le_compat; apply RRle_abs.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
Rabs (RiemannInt_SF (wn n)) +
Rabs (RiemannInt_SF (wn m)) < un n + un m
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
un n + un m < eps
  apply Rplus_lt_compat; assumption.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
un n + un m < eps
  apply Rle_lt_trans with (Rabs (un n) + Rabs (un m)).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
un n + un m <= Rabs (un n) + Rabs (un m)
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
Rabs (un n) + Rabs (un m) < eps
  apply Rplus_le_compat; apply RRle_abs.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> R_dist (un n0) 0 < eps0
H0:a <= b
H1:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
eps:R
H2:eps > 0
H3:0 < eps / 2
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 2
n, m:nat
H5:(n >= N0)%nat
H6:(m >= N0)%nat
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn m t) <= wn m t
H8:Rabs (RiemannInt_SF (wn m)) < un m
H9:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H10:Rabs (RiemannInt_SF (wn n)) < un n
Rabs (un n) + Rabs (un m) < eps
  replace (pos (un n)) with (un n - 0); [ idtac | ring ];
  replace (pos (un m)) with (un m - 0); [ idtac | ring ];
  rewrite (double_var eps); apply Rplus_lt_compat; apply H4;
    assumption.
Qed.

Lemma RiemannInt_P3 :
  forall (f:R -> R) (a b:R) (un:nat -> posreal) (vn wn:nat -> StepFun a b),
    Un_cv un 0 ->
    (forall n:nat,
      (forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\
      Rabs (RiemannInt_SF (wn n)) < un n) ->
    { l:R | Un_cv (fun N:nat => RiemannInt_SF (vn N)) l }.
forall (f : R -> R) (a b : R) (un : nat -> posreal)
  (vn wn : nat -> StepFun a b),
Un_cv (fun x : nat => un x) 0 ->
(forall n : nat,
 (forall t : R,
  Rmin a b <= t <= Rmax a b ->
  Rabs (f t - vn n t) <= wn n t) /\
 Rabs (RiemannInt_SF (wn n)) < un n) ->
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
Proof.
forall (f : R -> R) (a b : R) (un : nat -> posreal)
  (vn wn : nat -> StepFun a b),
Un_cv (fun x : nat => un x) 0 ->
(forall n : nat,
 (forall t : R,
  Rmin a b <= t <= Rmax a b ->
  Rabs (f t - vn n t) <= wn n t) /\
 Rabs (RiemannInt_SF (wn n)) < un n) ->
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  intros; destruct (Rle_dec a b) as [Hle|Hnle].
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hle:a <= b
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  apply RiemannInt_P2 with f un wn; assumption.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  assert (H1 : b <= a); auto with real.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  set (vn' := fun n:nat => mkStepFun (StepFun_P6 (pre (vn n))));
    set (wn' := fun n:nat => mkStepFun (StepFun_P6 (pre (wn n))));
      assert
        (H2 :
          forall n:nat,
            (forall t:R,
              Rmin b a <= t <= Rmax b a -> Rabs (f t - vn' n t) <= wn' n t) /\
            Rabs (RiemannInt_SF (wn' n)) < un n).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
H2:forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  intro; elim (H0 n); intros; split.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
forall t : R,
Rmin b a <= t <= Rmax b a ->
Rabs (f t - vn' n t) <= wn' n t
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Rabs (RiemannInt_SF (wn' n)) < un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
H2:forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  intros t (H4,H5); apply (H2 t); split;
    [ apply Rle_trans with (Rmin b a); try assumption; right;
      unfold Rmin
      | apply Rle_trans with (Rmax b a); try assumption; right;
        unfold Rmax ];
    decide (Rle_dec a b) with Hnle; decide (Rle_dec b a) with H1; reflexivity.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Rabs (RiemannInt_SF (wn' n)) < un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
H2:forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  generalize H3; unfold RiemannInt_SF; destruct (Rle_dec a b) as [Hleab|Hnleab];
    destruct (Rle_dec b a) as [Hle'|Hnle']; unfold wn'; intros;
      (replace
        (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (wn n)))))
          (subdivision (mkStepFun (StepFun_P6 (pre (wn n)))))) with
        (Int_SF (subdivision_val (wn n)) (subdivision (wn n)));
        [ idtac
          | apply StepFun_P17 with (fe (wn n)) a b;
            [ apply StepFun_P1
              | apply StepFun_P2;
                apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (wn n))))) ] ]).
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hleab:a <= b
Hle':b <= a
H4:Rabs
  (Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hleab:a <= b
Hnle':~ b <= a
H4:Rabs
  (Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hnleab:~ a <= b
Hle':b <= a
H4:Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hnleab:~ a <= b
Hnle':~ b <= a
H4:Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
H2:forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  apply H4.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hleab:a <= b
Hnle':~ b <= a
H4:Rabs
  (Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hnleab:~ a <= b
Hle':b <= a
H4:Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hnleab:~ a <= b
Hnle':~ b <= a
H4:Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
H2:forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  rewrite Rabs_Ropp; apply H4.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hnleab:~ a <= b
Hle':b <= a
H4:Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hnleab:~ a <= b
Hnle':~ b <= a
H4:Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
H2:forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  rewrite Rabs_Ropp in H4; apply H4.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
n:nat
H2:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - vn n t) <= wn n t
H3:Rabs (RiemannInt_SF (wn n)) < un n
Hnleab:~ a <= b
Hnle':~ b <= a
H4:Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
Rabs
  (-
   Int_SF (subdivision_val (wn n))
     (subdivision (wn n))) < 
un n
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
H2:forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  apply H4.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x : nat => un x) 0
H0:forall n : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n
Hnle:~ a <= b
H1:b <= a
vn':=fun n : nat =>
{| fe := vn n; pre := StepFun_P6 (pre (vn n)) |}:nat -> StepFun b a
wn':=fun n : nat =>
{| fe := wn n; pre := StepFun_P6 (pre (wn n)) |}:nat -> StepFun b a
H2:forall n : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n
{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn N)) l}
  assert (H3 := RiemannInt_P2 _ _ _ _ H H1 H2); elim H3; intros x p;
    exists (- x); unfold Un_cv; unfold Un_cv in p;
      intros; elim (p _ H4); intros; exists x0; intros;
        generalize (H5 _ H6); unfold R_dist, RiemannInt_SF;
          destruct (Rle_dec b a) as [Hle'|Hnle']; destruct (Rle_dec a b) as [Hle''|Hnle''];
          intros.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hle':b <= a
Hle'':a <= b
H7:Rabs
  (Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hle':b <= a
Hnle'':~ a <= b
H7:Rabs
  (Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (-
   Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hnle':~ b <= a
Hle'':a <= b
H7:Rabs
  (-
   Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hnle':~ b <= a
Hnle'':~ a <= b
H7:Rabs
  (-
   Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (-
   Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
  elim Hnle; assumption.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hle':b <= a
Hnle'':~ a <= b
H7:Rabs
  (Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (-
   Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hnle':~ b <= a
Hle'':a <= b
H7:Rabs
  (-
   Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hnle':~ b <= a
Hnle'':~ a <= b
H7:Rabs
  (-
   Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (-
   Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
  unfold vn' in H7;
    replace (Int_SF (subdivision_val (vn n)) (subdivision (vn n))) with
    (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (vn n)))))
      (subdivision (mkStepFun (StepFun_P6 (pre (vn n))))));
    [ unfold Rminus; rewrite Ropp_involutive; rewrite <- Rabs_Ropp;
      rewrite Ropp_plus_distr; rewrite Ropp_involutive;
        apply H7
      | symmetry ; apply StepFun_P17 with (fe (vn n)) a b;
        [ apply StepFun_P1
          | apply StepFun_P2;
            apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (vn n))))) ] ].
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hnle':~ b <= a
Hle'':a <= b
H7:Rabs
  (-
   Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hnle':~ b <= a
Hnle'':~ a <= b
H7:Rabs
  (-
   Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (-
   Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
  elim Hnle'; assumption.
f:R -> R
a, b:R
un:nat -> posreal
vn, wn:nat -> StepFun a b
H:Un_cv (fun x1 : nat => un x1) 0
H0:forall n0 : nat,
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - vn n0 t) <= wn n0 t) /\
Rabs (RiemannInt_SF (wn n0)) < un n0
Hnle:~ a <= b
H1:b <= a
vn':=fun n0 : nat =>
{|
fe := vn n0;
pre := StepFun_P6 (pre (vn n0)) |}:nat -> StepFun b a
wn':=fun n0 : nat =>
{|
fe := wn n0;
pre := StepFun_P6 (pre (wn n0)) |}:nat -> StepFun b a
H2:forall n0 : nat,
(forall t : R,
 Rmin b a <= t <= Rmax b a ->
 Rabs (f t - vn' n0 t) <= wn' n0 t) /\
Rabs (RiemannInt_SF (wn' n0)) < un n0
H3:{l : R |
Un_cv (fun N : nat => RiemannInt_SF (vn' N)) l}
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  R_dist (RiemannInt_SF (vn' n0)) x < eps0
eps:R
H4:eps > 0
x0:nat
H5:forall n0 : nat,
(n0 >= x0)%nat ->
R_dist (RiemannInt_SF (vn' n0)) x < eps
n:nat
H6:(n >= x0)%nat
Hnle':~ b <= a
Hnle'':~ a <= b
H7:Rabs
  (-
   Int_SF (subdivision_val (vn' n))
     (subdivision (vn' n)) - x) < eps
Rabs
  (-
   Int_SF (subdivision_val (vn n))
     (subdivision (vn n)) - 
   - x) < eps
  elim Hnle'; assumption.
Qed.

Lemma RiemannInt_exists :
  forall (f:R -> R) (a b:R) (pr:Riemann_integrable f a b)
    (un:nat -> posreal),
    Un_cv un 0 ->
    { l:R | Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr N)) l }.
forall (f : R -> R) (a b : R)
  (pr : Riemann_integrable f a b)
  (un : nat -> posreal),
Un_cv (fun x : nat => un x) 0 ->
{l : R |
Un_cv
  (fun N : nat => RiemannInt_SF (phi_sequence un pr N))
  l}
Proof.
forall (f : R -> R) (a b : R)
  (pr : Riemann_integrable f a b)
  (un : nat -> posreal),
Un_cv (fun x : nat => un x) 0 ->
{l : R |
Un_cv
  (fun N : nat => RiemannInt_SF (phi_sequence un pr N))
  l}
  intros f; intros;
    apply RiemannInt_P3 with
      f un (fun n:nat => proj1_sig (phi_sequence_prop un pr n));
      [ apply H | intro; apply (proj2_sig (phi_sequence_prop un pr n)) ].
Qed.

Lemma RiemannInt_P4 :
  forall (f:R -> R) (a b l:R) (pr1 pr2:Riemann_integrable f a b)
    (un vn:nat -> posreal),
    Un_cv un 0 ->
    Un_cv vn 0 ->
    Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr1 N)) l ->
    Un_cv (fun N:nat => RiemannInt_SF (phi_sequence vn pr2 N)) l.
forall (f : R -> R) (a b l : R)
  (pr1 pr2 : Riemann_integrable f a b)
  (un vn : nat -> posreal),
Un_cv (fun x : nat => un x) 0 ->
Un_cv (fun x : nat => vn x) 0 ->
Un_cv
  (fun N : nat =>
   RiemannInt_SF (phi_sequence un pr1 N)) l ->
Un_cv
  (fun N : nat =>
   RiemannInt_SF (phi_sequence vn pr2 N)) l
Proof.
forall (f : R -> R) (a b l : R)
  (pr1 pr2 : Riemann_integrable f a b)
  (un vn : nat -> posreal),
Un_cv (fun x : nat => un x) 0 ->
Un_cv (fun x : nat => vn x) 0 ->
Un_cv
  (fun N : nat =>
   RiemannInt_SF (phi_sequence un pr1 N)) l ->
Un_cv
  (fun N : nat =>
   RiemannInt_SF (phi_sequence vn pr2 N)) l
  unfold Un_cv; unfold R_dist; intros f; intros;
    assert (H3 : 0 < eps / 3).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (un n - 0) < eps0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (vn n - 0) < eps0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs
    (RiemannInt_SF (phi_sequence un pr1 n) - l) <
  eps0
eps:R
H2:eps > 0
0 < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (un n - 0) < eps0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (vn n - 0) < eps0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs
    (RiemannInt_SF (phi_sequence un pr1 n) - l) <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / 3
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (RiemannInt_SF (phi_sequence vn pr2 n) - l) <
  eps
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (un n - 0) < eps0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (vn n - 0) < eps0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs
    (RiemannInt_SF (phi_sequence un pr1 n) - l) <
  eps0
eps:R
H2:eps > 0
H3:0 < eps / 3
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (RiemannInt_SF (phi_sequence vn pr2 n) - l) <
  eps
  elim (H _ H3); clear H; intros N0 H; elim (H0 _ H3); clear H0; intros N1 H0;
    elim (H1 _ H3); clear H1; intros N2 H1; set (N := max (max N0 N1) N2);
      exists N; intros;
        apply Rle_lt_trans with
          (Rabs
            (RiemannInt_SF (phi_sequence vn pr2 n) -
              RiemannInt_SF (phi_sequence un pr1 n)) +
            Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence vn pr2 n) - l) <=
Rabs
  (RiemannInt_SF (phi_sequence vn pr2 n) -
   RiemannInt_SF (phi_sequence un pr1 n)) +
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs
  (RiemannInt_SF (phi_sequence vn pr2 n) -
   RiemannInt_SF (phi_sequence un pr1 n)) +
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) < eps
  replace (RiemannInt_SF (phi_sequence vn pr2 n) - l) with
  (RiemannInt_SF (phi_sequence vn pr2 n) -
    RiemannInt_SF (phi_sequence un pr1 n) +
    (RiemannInt_SF (phi_sequence un pr1 n) - l)); [ apply Rabs_triang | ring ].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs
  (RiemannInt_SF (phi_sequence vn pr2 n) -
   RiemannInt_SF (phi_sequence un pr1 n)) +
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) < eps
  replace eps with (2 * (eps / 3) + eps / 3).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs
  (RiemannInt_SF (phi_sequence vn pr2 n) -
   RiemannInt_SF (phi_sequence un pr1 n)) +
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
2 * (eps / 3) + eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rplus_lt_compat.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs
  (RiemannInt_SF (phi_sequence vn pr2 n) -
   RiemannInt_SF (phi_sequence un pr1 n)) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  elim (phi_sequence_prop vn pr2 n); intros psi_vn H5;
    elim (phi_sequence_prop un pr1 n); intros psi_un H6;
      replace
      (RiemannInt_SF (phi_sequence vn pr2 n) -
        RiemannInt_SF (phi_sequence un pr1 n)) with
      (RiemannInt_SF (phi_sequence vn pr2 n) +
        -1 * RiemannInt_SF (phi_sequence un pr1 n)); [ idtac | ring ];
      rewrite <- StepFun_P30.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  destruct (Rle_dec a b) as [Hle|Hnle].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rle_lt_trans with
    (RiemannInt_SF
      (mkStepFun
        (StepFun_P32
          (mkStepFun
            (StepFun_P28 (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n)))))).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <=
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := fun x0 : R =>
                 phi_sequence vn pr2 n x0 +
                 -1 * phi_sequence un pr1 n x0;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |} x);
  pre := StepFun_P32
           {|
           fe := fun x : R =>
                 phi_sequence vn pr2 n x +
                 -1 * phi_sequence un pr1 n x;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |} |}
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := fun x0 : R =>
                 phi_sequence vn pr2 n x0 +
                 -1 * phi_sequence un pr1 n x0;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |} x);
  pre := StepFun_P32
           {|
           fe := fun x : R =>
                 phi_sequence vn pr2 n x +
                 -1 * phi_sequence un pr1 n x;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |} |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply StepFun_P34; assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := fun x0 : R =>
                 phi_sequence vn pr2 n x0 +
                 -1 * phi_sequence un pr1 n x0;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |} x);
  pre := StepFun_P32
           {|
           fe := fun x : R =>
                 phi_sequence vn pr2 n x +
                 -1 * phi_sequence un pr1 n x;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |} |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rle_lt_trans with
    (RiemannInt_SF (mkStepFun (StepFun_P28 1 psi_un psi_vn))).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := fun x0 : R =>
                 phi_sequence vn pr2 n x0 +
                 -1 * phi_sequence un pr1 n x0;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |} x);
  pre := StepFun_P32
           {|
           fe := fun x : R =>
                 phi_sequence vn pr2 n x +
                 -1 * phi_sequence un pr1 n x;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |} |} <=
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |}
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply StepFun_P37; try assumption; intros; simpl; rewrite Rmult_1_l;
    apply Rle_trans with
      (Rabs (phi_sequence vn pr2 n x - f x) +
        Rabs (f x - phi_sequence un pr1 n x)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
Rabs
  (phi_sequence vn pr2 n x +
   -1 * phi_sequence un pr1 n x) <=
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_un x + psi_vn x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  replace (phi_sequence vn pr2 n x + -1 * phi_sequence un pr1 n x) with
  (phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x));
  [ apply Rabs_triang | ring ].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_un x + psi_vn x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  assert (H10 : Rmin a b = a).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
Rmin a b = a
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_un x + psi_vn x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  unfold Rmin; decide (Rle_dec a b) with Hle; reflexivity.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_un x + psi_vn x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  assert (H11 : Rmax a b = b).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
Rmax a b = b
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
H11:Rmax a b = b
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_un x + psi_vn x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  unfold Rmax; decide (Rle_dec a b) with Hle; reflexivity.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
H11:Rmax a b = b
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_un x + psi_vn x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite (Rplus_comm (psi_un x)); apply Rplus_le_compat.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
H11:Rmax a b = b
Rabs (phi_sequence vn pr2 n x - f x) <= psi_vn x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
H11:Rmax a b = b
Rabs (f x - phi_sequence un pr1 n x) <= psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; destruct H5 as (H8,H9); apply H8.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H8:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t
H9:Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
H11:Rmax a b = b
Rmin a b <= x <= Rmax a b
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
H11:Rmax a b = b
Rabs (f x - phi_sequence un pr1 n x) <= psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite H10; rewrite H11; elim H7; intros; split; left; assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
H11:Rmax a b = b
Rabs (f x - phi_sequence un pr1 n x) <= psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  elim H6; intros; apply H8.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
x:R
H7:a < x < b
H10:Rmin a b = a
H11:Rmax a b = b
H8:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr1 n t) <= psi_un t
H9:Rabs (RiemannInt_SF psi_un) < un n
Rmin a b <= x <= Rmax a b
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite H10; rewrite H11; elim H7; intros; split; left; assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF
  {|
  fe := fun x : R => psi_un x + 1 * psi_vn x;
  pre := StepFun_P28 1 psi_un psi_vn |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_un < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_vn < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rlt_trans with (pos (un n)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_un < un n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
un n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_vn < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  elim H6; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_un)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr1 n t) <= psi_un t
H8:Rabs (RiemannInt_SF psi_un) < un n
RiemannInt_SF psi_un <= Rabs (RiemannInt_SF psi_un)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr1 n t) <= psi_un t
H8:Rabs (RiemannInt_SF psi_un) < un n
Rabs (RiemannInt_SF psi_un) < un n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
un n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_vn < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply RRle_abs.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr1 n t) <= psi_un t
H8:Rabs (RiemannInt_SF psi_un) < un n
Rabs (RiemannInt_SF psi_un) < un n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
un n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_vn < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
un n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_vn < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  replace (pos (un n)) with (Rabs (un n - 0));
  [ apply H; unfold ge; apply le_trans with N; try assumption;
    unfold N; apply le_trans with (max N0 N1);
      apply le_max_l
    | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
      apply Rle_ge; left; apply (cond_pos (un n)) ].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_vn < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rlt_trans with (pos (vn n)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
RiemannInt_SF psi_vn < vn n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
vn n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  elim H5; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_vn)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t
H8:Rabs (RiemannInt_SF psi_vn) < vn n
RiemannInt_SF psi_vn <= Rabs (RiemannInt_SF psi_vn)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t
H8:Rabs (RiemannInt_SF psi_vn) < vn n
Rabs (RiemannInt_SF psi_vn) < vn n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
vn n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply RRle_abs; assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t
H8:Rabs (RiemannInt_SF psi_vn) < vn n
Rabs (RiemannInt_SF psi_vn) < vn n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
vn n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hle:a <= b
vn n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  replace (pos (vn n)) with (Rabs (vn n - 0));
  [ apply H0; unfold ge; apply le_trans with N; try assumption;
    unfold N; apply le_trans with (max N0 N1);
      [ apply le_max_r | apply le_max_l ]
    | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
      apply Rle_ge; left; apply (cond_pos (vn n)) ].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := fun x : R =>
           phi_sequence vn pr2 n x +
           -1 * phi_sequence un pr1 n x;
     pre := StepFun_P28 
              (-1) (phi_sequence vn pr2 n)
              (phi_sequence un pr1 n) |}) <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite StepFun_P39; rewrite Rabs_Ropp;
    apply Rle_lt_trans with
      (RiemannInt_SF
        (mkStepFun
          (StepFun_P32
            (mkStepFun
              (StepFun_P6
                (pre
                  (mkStepFun
                    (StepFun_P28 (-1) (phi_sequence vn pr2 n)
                      (phi_sequence un pr1 n))))))))).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
Rabs
  (RiemannInt_SF
     {|
     fe := {|
           fe := fun x : R =>
                 phi_sequence vn pr2 n x +
                 -1 * phi_sequence un pr1 n x;
           pre := StepFun_P28 
                    (-1) 
                    (phi_sequence vn pr2 n)
                    (phi_sequence un pr1 n) |};
     pre := StepFun_P6
              (pre
                 {|
                 fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |}) <=
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := {|
                 fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
           pre := StepFun_P6
                    (pre
                       {|
                       fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                       pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |}
             x);
  pre := StepFun_P32
           {|
           fe := {|
                 fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
           pre := StepFun_P6
                    (pre
                       {|
                       fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                       pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |} |}
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := {|
                 fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
           pre := StepFun_P6
                    (pre
                       {|
                       fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                       pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |}
             x);
  pre := StepFun_P32
           {|
           fe := {|
                 fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
           pre := StepFun_P6
                    (pre
                       {|
                       fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                       pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |} |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply StepFun_P34; try auto with real.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := {|
                 fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
           pre := StepFun_P6
                    (pre
                       {|
                       fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                       pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |}
             x);
  pre := StepFun_P32
           {|
           fe := {|
                 fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
           pre := StepFun_P6
                    (pre
                       {|
                       fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                       pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |} |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rle_lt_trans with
    (RiemannInt_SF
      (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 psi_vn psi_un)))))).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := fun x : R =>
        Rabs
          ({|
           fe := {|
                 fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
           pre := StepFun_P6
                    (pre
                       {|
                       fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                       pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |}
             x);
  pre := StepFun_P32
           {|
           fe := {|
                 fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                 pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
           pre := StepFun_P6
                    (pre
                       {|
                       fe := fun x : R =>
                       phi_sequence vn pr2 n x +
                       -1 * phi_sequence un pr1 n x;
                       pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |} |} <=
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |}
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply StepFun_P37.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
b <= a
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
forall x : R,
b < x < a ->
{|
fe := fun x0 : R =>
      Rabs
        ({|
         fe := {|
               fe := fun x1 : R =>
                     phi_sequence vn pr2 n x1 +
                     -1 * phi_sequence un pr1 n x1;
               pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
         pre := StepFun_P6
                  (pre
                     {|
                     fe := fun x1 : R =>
                       phi_sequence vn pr2 n x1 +
                       -1 * phi_sequence un pr1 n x1;
                     pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |}
           x0);
pre := StepFun_P32
         {|
         fe := {|
               fe := fun x0 : R =>
                     phi_sequence vn pr2 n x0 +
                     -1 * phi_sequence un pr1 n x0;
               pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
         pre := StepFun_P6
                  (pre
                     {|
                     fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                     pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |} |}
  x <=
{|
fe := {|
      fe := fun x0 : R => psi_vn x0 + 1 * psi_un x0;
      pre := StepFun_P28 1 psi_vn psi_un |};
pre := StepFun_P6
         (pre
            {|
            fe := fun x0 : R =>
                  psi_vn x0 + 1 * psi_un x0;
            pre := StepFun_P28 1 psi_vn psi_un |}) |}
  x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  auto with real.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
forall x : R,
b < x < a ->
{|
fe := fun x0 : R =>
      Rabs
        ({|
         fe := {|
               fe := fun x1 : R =>
                     phi_sequence vn pr2 n x1 +
                     -1 * phi_sequence un pr1 n x1;
               pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
         pre := StepFun_P6
                  (pre
                     {|
                     fe := fun x1 : R =>
                       phi_sequence vn pr2 n x1 +
                       -1 * phi_sequence un pr1 n x1;
                     pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |}
           x0);
pre := StepFun_P32
         {|
         fe := {|
               fe := fun x0 : R =>
                     phi_sequence vn pr2 n x0 +
                     -1 * phi_sequence un pr1 n x0;
               pre := StepFun_P28 
                       (-1) 
                       (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |};
         pre := StepFun_P6
                  (pre
                     {|
                     fe := fun x0 : R =>
                       phi_sequence vn pr2 n x0 +
                       -1 * phi_sequence un pr1 n x0;
                     pre := StepFun_P28 
                       (-1) (phi_sequence vn pr2 n)
                       (phi_sequence un pr1 n) |}) |} |}
  x <=
{|
fe := {|
      fe := fun x0 : R => psi_vn x0 + 1 * psi_un x0;
      pre := StepFun_P28 1 psi_vn psi_un |};
pre := StepFun_P6
         (pre
            {|
            fe := fun x0 : R =>
                  psi_vn x0 + 1 * psi_un x0;
            pre := StepFun_P28 1 psi_vn psi_un |}) |}
  x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  intros; simpl; rewrite Rmult_1_l;
    apply Rle_trans with
      (Rabs (phi_sequence vn pr2 n x - f x) +
        Rabs (f x - phi_sequence un pr1 n x)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
Rabs
  (phi_sequence vn pr2 n x +
   -1 * phi_sequence un pr1 n x) <=
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_vn x + psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  replace (phi_sequence vn pr2 n x + -1 * phi_sequence un pr1 n x) with
  (phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x));
  [ apply Rabs_triang | ring ].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_vn x + psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  assert (H10 : Rmin a b = b).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
Rmin a b = b
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_vn x + psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  unfold Rmin; decide (Rle_dec a b) with Hnle; reflexivity.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_vn x + psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  assert (H11 : Rmax a b = a).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
Rmax a b = a
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
H11:Rmax a b = a
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_vn x + psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  unfold Rmax; decide (Rle_dec a b) with Hnle; reflexivity.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
H11:Rmax a b = a
Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x) <=
psi_vn x + psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rplus_le_compat.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
H11:Rmax a b = a
Rabs (phi_sequence vn pr2 n x - f x) <= psi_vn x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
H11:Rmax a b = a
Rabs (f x - phi_sequence un pr1 n x) <= psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
H11:Rmax a b = a
H8:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t
H9:Rabs (RiemannInt_SF psi_vn) < vn n
Rmin a b <= x <= Rmax a b
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
H11:Rmax a b = a
Rabs (f x - phi_sequence un pr1 n x) <= psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite H10; rewrite H11; elim H7; intros; split; left; assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
H11:Rmax a b = a
Rabs (f x - phi_sequence un pr1 n x) <= psi_un x
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  elim H6; intros; apply H8.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
x:R
H7:b < x < a
H10:Rmin a b = b
H11:Rmax a b = a
H8:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr1 n t) <= psi_un t
H9:Rabs (RiemannInt_SF psi_un) < un n
Rmin a b <= x <= Rmax a b
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite H10; rewrite H11; elim H7; intros; split; left; assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
RiemannInt_SF
  {|
  fe := {|
        fe := fun x : R => psi_vn x + 1 * psi_un x;
        pre := StepFun_P28 1 psi_vn psi_un |};
  pre := StepFun_P6
           (pre
              {|
              fe := fun x : R =>
                    psi_vn x + 1 * psi_un x;
              pre := StepFun_P28 1 psi_vn psi_un |}) |} <
2 * (eps / 3)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite <-
    (Ropp_involutive
      (RiemannInt_SF
        (mkStepFun
          (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 psi_vn psi_un)))))))
    ; rewrite <- StepFun_P39; rewrite StepFun_P30; rewrite Rmult_1_l;
      rewrite double; rewrite Ropp_plus_distr; apply Rplus_lt_compat.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_vn < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_un < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rlt_trans with (pos (vn n)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_vn < vn n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
vn n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_un < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  elim H5; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_vn)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t
H8:Rabs (RiemannInt_SF psi_vn) < vn n
- RiemannInt_SF psi_vn <= Rabs (RiemannInt_SF psi_vn)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t
H8:Rabs (RiemannInt_SF psi_vn) < vn n
Rabs (RiemannInt_SF psi_vn) < vn n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
vn n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_un < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite <- Rabs_Ropp; apply RRle_abs.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t
H8:Rabs (RiemannInt_SF psi_vn) < vn n
Rabs (RiemannInt_SF psi_vn) < vn n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
vn n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_un < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
vn n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_un < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  replace (pos (vn n)) with (Rabs (vn n - 0));
  [ apply H0; unfold ge; apply le_trans with N; try assumption;
    unfold N; apply le_trans with (max N0 N1);
      [ apply le_max_r | apply le_max_l ]
    | unfold R_dist; unfold Rminus; rewrite Ropp_0;
      rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
        left; apply (cond_pos (vn n)) ].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_un < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rlt_trans with (pos (un n)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
- RiemannInt_SF psi_un < un n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
un n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  elim H6; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_un)).
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr1 n t) <= psi_un t
H8:Rabs (RiemannInt_SF psi_un) < un n
- RiemannInt_SF psi_un <= Rabs (RiemannInt_SF psi_un)
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr1 n t) <= psi_un t
H8:Rabs (RiemannInt_SF psi_un) < un n
Rabs (RiemannInt_SF psi_un) < un n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
un n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  rewrite <- Rabs_Ropp; apply RRle_abs; assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
H7:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr1 n t) <= psi_un t
H8:Rabs (RiemannInt_SF psi_un) < un n
Rabs (RiemannInt_SF psi_un) < un n
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
un n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  assumption.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
psi_vn:StepFun a b
H5:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence vn pr2 n t) <= psi_vn t) /\
Rabs (RiemannInt_SF psi_vn) < vn n
psi_un:StepFun a b
H6:(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (f t - phi_sequence un pr1 n t) <= psi_un t) /\
Rabs (RiemannInt_SF psi_un) < un n
Hnle:~ a <= b
un n < eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  replace (pos (un n)) with (Rabs (un n - 0));
  [ apply H; unfold ge; apply le_trans with N; try assumption;
    unfold N; apply le_trans with (max N0 N1);
      apply le_max_l
    | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
      apply Rle_ge; left; apply (cond_pos (un n)) ].
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l) <
eps / 3
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply H1; unfold ge; apply le_trans with N; try assumption;
    unfold N; apply le_max_r.
f:R -> R
a, b, l:R
pr1, pr2:Riemann_integrable f a b
un, vn:nat -> posreal
eps:R
H2:eps > 0
H3:0 < eps / 3
N0:nat
H:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (un n0 - 0) < eps / 3
N1:nat
H0:forall n0 : nat,
(n0 >= N1)%nat -> Rabs (vn n0 - 0) < eps / 3
N2:nat
H1:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (RiemannInt_SF (phi_sequence un pr1 n0) - l) <
eps / 3
N:=Nat.max (Nat.max N0 N1) N2:nat
n:nat
H4:(n >= N)%nat
2 * (eps / 3) + eps / 3 = eps
  apply Rmult_eq_reg_l with 3;
    [ unfold Rdiv; rewrite Rmult_plus_distr_l;
      do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
        rewrite <- Rinv_l_sym; [ ring | discrR ]
      | discrR ].
Qed.

Lemma RinvN_pos : forall n:nat, 0 < / (INR n + 1).
forall n : nat, 0 < / (INR n + 1)
Proof.
forall n : nat, 0 < / (INR n + 1)
  intro; apply Rinv_0_lt_compat; apply Rplus_le_lt_0_compat;
    [ apply pos_INR | apply Rlt_0_1 ].
Qed.

Definition RinvN (N:nat) : posreal := mkposreal _ (RinvN_pos N).

Lemma RinvN_cv : Un_cv RinvN 0.
Un_cv (fun N : nat => RinvN N) 0
Proof.
Un_cv (fun N : nat => RinvN N) 0
  unfold Un_cv; intros; assert (H0 := archimed (/ eps)); elim H0;
    clear H0; intros; assert (H2 : (0 <= up (/ eps))%Z).
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
(0 <= up (/ eps))%Z
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (RinvN n) 0 < eps
  apply le_IZR; left; apply Rlt_trans with (/ eps);
    [ apply Rinv_0_lt_compat; assumption | assumption ].
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (RinvN n) 0 < eps
  elim (IZN _ H2); intros; exists x; intros; unfold R_dist;
    simpl; unfold Rminus; rewrite Ropp_0;
      rewrite Rplus_0_r; assert (H5 : 0 < INR n + 1).
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
0 < INR n + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
Rabs (/ (INR n + 1)) < eps
  apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
Rabs (/ (INR n + 1)) < eps
  rewrite Rabs_right;
    [ idtac
      | left; change (0 < / (INR n + 1)); apply Rinv_0_lt_compat;
        assumption ]; apply Rle_lt_trans with (/ (INR x + 1)).
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ (INR n + 1) <= / (INR x + 1)
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ (INR x + 1) < eps
  apply Rinv_le_contravar.
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
0 < INR x + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
INR x + 1 <= INR n + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ (INR x + 1) < eps
  apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
INR x + 1 <= INR n + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ (INR x + 1) < eps
  apply Rplus_le_compat_r; apply le_INR; apply H4.
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ (INR x + 1) < eps
  rewrite <- (Rinv_involutive eps).
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ (INR x + 1) < / / eps
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
eps <> 0
  apply Rinv_lt_contravar.
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
0 < / eps * (INR x + 1)
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ eps < INR x + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
eps <> 0
  apply Rmult_lt_0_compat.
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
0 < / eps
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
0 < INR x + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ eps < INR x + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
eps <> 0
  apply Rinv_0_lt_compat; assumption.
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
0 < INR x + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ eps < INR x + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
eps <> 0
  apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
/ eps < INR x + 1
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
eps <> 0
  apply Rlt_trans with (INR x);
    [ rewrite INR_IZR_INZ; rewrite <- H3; apply H0
      | pattern (INR x) at 1; rewrite <- Rplus_0_r;
        apply Rplus_lt_compat_l; apply Rlt_0_1 ].
eps:R
H:eps > 0
H0:IZR (up (/ eps)) > / eps
H1:IZR (up (/ eps)) - / eps <= 1
H2:(0 <= up (/ eps))%Z
x:nat
H3:up (/ eps) = Z.of_nat x
n:nat
H4:(n >= x)%nat
H5:0 < INR n + 1
eps <> 0
  red; intro; rewrite H6 in H; elim (Rlt_irrefl _ H).
Qed.

Lemma Riemann_integrable_ext :
  forall f g a b, 
    (forall x, Rmin a b <= x <= Rmax a b -> f x = g x) ->
    Riemann_integrable f a b -> Riemann_integrable g a b.
forall (f g : R -> R) (a b : R),
(forall x : R, Rmin a b <= x <= Rmax a b -> f x = g x) ->
Riemann_integrable f a b -> Riemann_integrable g a b
intros f g a b fg rif eps; destruct (rif eps) as [phi [psi [P1 P2]]].
f, g:R -> R
a, b:R
fg:forall x : R,
Rmin a b <= x <= Rmax a b -> f x = g x
rif:Riemann_integrable f a b
eps:posreal
phi, psi:StepFun a b
P1:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi t) <= psi t
P2:Rabs (RiemannInt_SF psi) < eps
{phi0 : StepFun a b &
{psi0 : StepFun a b |
(forall t : R,
 Rmin a b <= t <= Rmax a b ->
 Rabs (g t - phi0 t) <= psi0 t) /\
Rabs (RiemannInt_SF psi0) < eps}}
exists phi; exists psi;split;[ | assumption ].
f, g:R -> R
a, b:R
fg:forall x : R,
Rmin a b <= x <= Rmax a b -> f x = g x
rif:Riemann_integrable f a b
eps:posreal
phi, psi:StepFun a b
P1:forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi t) <= psi t
P2:Rabs (RiemannInt_SF psi) < eps
forall t : R,
Rmin a b <= t <= Rmax a b ->
Rabs (g t - phi t) <= psi t
intros t intt; rewrite <- fg;[ | assumption].
f, g:R -> R
a, b:R
fg:forall x : R,
Rmin a b <= x <= Rmax a b -> f x = g x
rif:Riemann_integrable f a b
eps:posreal
phi, psi:StepFun a b
P1:forall t0 : R,
Rmin a b <= t0 <= Rmax a b ->
Rabs (f t0 - phi t0) <= psi t0
P2:Rabs (RiemannInt_SF psi) < eps
t:R
intt:Rmin a b <= t <= Rmax a b
Rabs (f t - phi t) <= psi t
apply P1; assumption.
Qed.
(**********)
Definition RiemannInt (f:R -> R) (a b:R) (pr:Riemann_integrable f a b) : R :=
  let (a,_) := RiemannInt_exists pr RinvN RinvN_cv in a.

Lemma RiemannInt_P5 :
  forall (f:R -> R) (a b:R) (pr1 pr2:Riemann_integrable f a b),
    RiemannInt pr1 = RiemannInt pr2.
forall (f : R -> R) (a b : R)
  (pr1 pr2 : Riemann_integrable f a b),
RiemannInt pr1 = RiemannInt pr2
Proof.
forall (f : R -> R) (a b : R)
  (pr1 pr2 : Riemann_integrable f a b),
RiemannInt pr1 = RiemannInt pr2
  intros; unfold RiemannInt;
    case (RiemannInt_exists pr1 RinvN RinvN_cv) as (x,HUn);
    case (RiemannInt_exists pr2 RinvN RinvN_cv) as (x0,HUn0);
      eapply UL_sequence;
        [ apply HUn
          | apply RiemannInt_P4 with pr2 RinvN; apply RinvN_cv || assumption ].
Qed.

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