Rsqrt_def.v

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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
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Require Import Sumbool.
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Ranalysis1.
Local Open Scope R_scope.

Fixpoint Dichotomy_lb (x y:R) (P:R -> bool) (N:nat) {struct N} : R :=
  match N with
    | O => x
    | S n =>
      let down := Dichotomy_lb x y P n in
        let up := Dichotomy_ub x y P n in
          let z := (down + up) / 2 in if P z then down else z
  end

  with Dichotomy_ub (x y:R) (P:R -> bool) (N:nat) {struct N} : R :=
    match N with
      | O => y
      | S n =>
        let down := Dichotomy_lb x y P n in
          let up := Dichotomy_ub x y P n in
            let z := (down + up) / 2 in if P z then z else up
    end.

Definition dicho_lb (x y:R) (P:R -> bool) (N:nat) : R := Dichotomy_lb x y P N.
Definition dicho_up (x y:R) (P:R -> bool) (N:nat) : R := Dichotomy_ub x y P N.

(**********)
Lemma dicho_comp :
  forall (x y:R) (P:R -> bool) (n:nat),
    x <= y -> dicho_lb x y P n <= dicho_up x y P n.forall (x y : R) (P : R -> bool) (n : nat),
x <= y -> dicho_lb x y P n <= dicho_up x y P n
Proof.forall (x y : R) (P : R -> bool) (n : nat),
x <= y -> dicho_lb x y P n <= dicho_up x y P n
  intros.x, y:R
P:R -> bool
n:nat
H:x <= y
dicho_lb x y P n <= dicho_up x y P n
  induction  n as [| n Hrecn].x, y:R
P:R -> bool
H:x <= y
dicho_lb x y P 0 <= dicho_up x y P 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
dicho_lb x y P (S n) <= dicho_up x y P (S n)
  simpl; assumption.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
dicho_lb x y P (S n) <= dicho_up x y P (S n)
  simpl.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(if
  P
    ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
 then Dichotomy_lb x y P n
 else
  (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) <=
(if
  P
    ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
 then
  (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
 else Dichotomy_ub x y P n)
  case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
Dichotomy_lb x y P n <=
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  unfold Rdiv; apply Rmult_le_reg_l with 2.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
0 < 2
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
2 * Dichotomy_lb x y P n <=
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  prove_sup0.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
2 * Dichotomy_lb x y P n <=
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  pattern 2 at 1; rewrite Rmult_comm.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
2 * Dichotomy_lb x y P n <=
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 *
2
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
2 * Dichotomy_lb x y P n <=
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * 1
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  rewrite Rmult_1_r.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
2 * Dichotomy_lb x y P n <=
Dichotomy_lb x y P n + Dichotomy_ub x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  rewrite double.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
Dichotomy_lb x y P n + Dichotomy_lb x y P n <=
Dichotomy_lb x y P n + Dichotomy_ub x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  apply Rplus_le_compat_l.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
Dichotomy_lb x y P n <= Dichotomy_ub x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  assumption.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
Dichotomy_ub x y P n
  unfold Rdiv; apply Rmult_le_reg_l with 2.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
0 < 2
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2) <=
2 * Dichotomy_ub x y P n
  prove_sup0.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2) <=
2 * Dichotomy_ub x y P n
  rewrite Rmult_comm.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 *
2 <= 2 * Dichotomy_ub x y P n
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * 1 <=
2 * Dichotomy_ub x y P n
  rewrite Rmult_1_r.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
Dichotomy_lb x y P n + Dichotomy_ub x y P n <=
2 * Dichotomy_ub x y P n
  rewrite double.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
Dichotomy_lb x y P n + Dichotomy_ub x y P n <=
Dichotomy_ub x y P n + Dichotomy_ub x y P n
  rewrite <- (Rplus_comm (Dichotomy_ub x y P n)).x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
Dichotomy_ub x y P n + Dichotomy_lb x y P n <=
Dichotomy_ub x y P n + Dichotomy_ub x y P n
  apply Rplus_le_compat_l.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_lb x y P n <= dicho_up x y P n
Dichotomy_lb x y P n <= Dichotomy_ub x y P n
  assumption.
Qed.

Lemma dicho_lb_growing :
  forall (x y:R) (P:R -> bool), x <= y -> Un_growing (dicho_lb x y P).forall (x y : R) (P : R -> bool),
x <= y -> Un_growing (dicho_lb x y P)
Proof.forall (x y : R) (P : R -> bool),
x <= y -> Un_growing (dicho_lb x y P)
  intros.x, y:R
P:R -> bool
H:x <= y
Un_growing (dicho_lb x y P)
  unfold Un_growing.x, y:R
P:R -> bool
H:x <= y
forall n : nat,
dicho_lb x y P n <= dicho_lb x y P (S n)
  intro.x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n <= dicho_lb x y P (S n)
  simpl.x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n <=
(if
  P
    ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
 then Dichotomy_lb x y P n
 else
  (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
  case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n <= Dichotomy_lb x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n <=
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  right; reflexivity.x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n <=
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  unfold Rdiv; apply Rmult_le_reg_l with 2.x, y:R
P:R -> bool
H:x <= y
n:nat
0 < 2
x, y:R
P:R -> bool
H:x <= y
n:nat
2 * dicho_lb x y P n <=
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2)
  prove_sup0.x, y:R
P:R -> bool
H:x <= y
n:nat
2 * dicho_lb x y P n <=
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2)
  pattern 2 at 1; rewrite Rmult_comm.x, y:R
P:R -> bool
H:x <= y
n:nat
2 * dicho_lb x y P n <=
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 *
2
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].x, y:R
P:R -> bool
H:x <= y
n:nat
2 * dicho_lb x y P n <=
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * 1
  rewrite Rmult_1_r.x, y:R
P:R -> bool
H:x <= y
n:nat
2 * dicho_lb x y P n <=
Dichotomy_lb x y P n + Dichotomy_ub x y P n
  rewrite double.x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n + dicho_lb x y P n <=
Dichotomy_lb x y P n + Dichotomy_ub x y P n
  apply Rplus_le_compat_l.x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n <= Dichotomy_ub x y P n
  replace (Dichotomy_ub x y P n) with (dicho_up x y P n);
  [ apply dicho_comp; assumption | reflexivity ].
Qed.

Lemma dicho_up_decreasing :
  forall (x y:R) (P:R -> bool), x <= y -> Un_decreasing (dicho_up x y P).forall (x y : R) (P : R -> bool),
x <= y -> Un_decreasing (dicho_up x y P)
Proof.forall (x y : R) (P : R -> bool),
x <= y -> Un_decreasing (dicho_up x y P)
  intros.x, y:R
P:R -> bool
H:x <= y
Un_decreasing (dicho_up x y P)
  unfold Un_decreasing.x, y:R
P:R -> bool
H:x <= y
forall n : nat,
dicho_up x y P (S n) <= dicho_up x y P n
  intro.x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_up x y P (S n) <= dicho_up x y P n
  simpl.x, y:R
P:R -> bool
H:x <= y
n:nat
(if
  P
    ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
 then
  (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
 else Dichotomy_ub x y P n) <= dicho_up x y P n
  case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).x, y:R
P:R -> bool
H:x <= y
n:nat
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <=
dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  unfold Rdiv; apply Rmult_le_reg_l with 2.x, y:R
P:R -> bool
H:x <= y
n:nat
0 < 2
x, y:R
P:R -> bool
H:x <= y
n:nat
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2) <=
2 * dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  prove_sup0.x, y:R
P:R -> bool
H:x <= y
n:nat
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2) <=
2 * dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  rewrite Rmult_comm.x, y:R
P:R -> bool
H:x <= y
n:nat
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 *
2 <= 2 * dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].x, y:R
P:R -> bool
H:x <= y
n:nat
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * 1 <=
2 * dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  rewrite Rmult_1_r.x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_lb x y P n + Dichotomy_ub x y P n <=
2 * dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  rewrite double.x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_lb x y P n + Dichotomy_ub x y P n <=
dicho_up x y P n + dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  replace (Dichotomy_ub x y P n) with (dicho_up x y P n);
  [ idtac | reflexivity ].x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_lb x y P n + dicho_up x y P n <=
dicho_up x y P n + dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  replace (Dichotomy_lb x y P n) with (dicho_lb x y P n);
  [ idtac | reflexivity ].x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n + dicho_up x y P n <=
dicho_up x y P n + dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  rewrite <- (Rplus_comm (dicho_up x y P n)).x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_up x y P n + dicho_lb x y P n <=
dicho_up x y P n + dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  apply Rplus_le_compat_l.x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n <= dicho_up x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  apply dicho_comp; assumption.x, y:R
P:R -> bool
H:x <= y
n:nat
Dichotomy_ub x y P n <= dicho_up x y P n
  right; reflexivity.
Qed.

Lemma dicho_lb_maj_y :
  forall (x y:R) (P:R -> bool), x <= y -> forall n:nat, dicho_lb x y P n <= y.forall (x y : R) (P : R -> bool),
x <= y -> forall n : nat, dicho_lb x y P n <= y
Proof.forall (x y : R) (P : R -> bool),
x <= y -> forall n : nat, dicho_lb x y P n <= y
  intros.x, y:R
P:R -> bool
H:x <= y
n:nat
dicho_lb x y P n <= y
  induction  n as [| n Hrecn].x, y:R
P:R -> bool
H:x <= y
dicho_lb x y P 0 <= y
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
dicho_lb x y P (S n) <= y
  simpl; assumption.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
dicho_lb x y P (S n) <= y
  simpl.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
(if
  P
    ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
 then Dichotomy_lb x y P n
 else
  (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) <=
y
  case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
Dichotomy_lb x y P n <= y
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <= y
  assumption.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 <= y
  unfold Rdiv; apply Rmult_le_reg_l with 2.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
0 < 2
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2) <=
2 * y
  prove_sup0.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2) <=
2 * y
  rewrite Rmult_comm.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 *
2 <= 2 * y
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ].x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
Dichotomy_lb x y P n + Dichotomy_ub x y P n <= 2 * y
  rewrite double; apply Rplus_le_compat.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
Dichotomy_lb x y P n <= y
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
Dichotomy_ub x y P n <= y
  assumption.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
Dichotomy_ub x y P n <= y
  pattern y at 2; replace y with (Dichotomy_ub x y P 0);
    [ idtac | reflexivity ].x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
Dichotomy_ub x y P n <= Dichotomy_ub x y P 0
  apply decreasing_prop.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
Un_decreasing (Dichotomy_ub x y P)
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
(0 <= n)%nat
  assert (H0 := dicho_up_decreasing x y P H).x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
H0:Un_decreasing (dicho_up x y P)
Un_decreasing (Dichotomy_ub x y P)
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
(0 <= n)%nat
  assumption.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:dicho_lb x y P n <= y
(0 <= n)%nat
  apply le_O_n.
Qed.

Lemma dicho_lb_maj :
  forall (x y:R) (P:R -> bool), x <= y -> has_ub (dicho_lb x y P).forall (x y : R) (P : R -> bool),
x <= y -> has_ub (dicho_lb x y P)
Proof.forall (x y : R) (P : R -> bool),
x <= y -> has_ub (dicho_lb x y P)
  intros.x, y:R
P:R -> bool
H:x <= y
has_ub (dicho_lb x y P)
  cut (forall n:nat, dicho_lb x y P n <= y).x, y:R
P:R -> bool
H:x <= y
(forall n : nat, dicho_lb x y P n <= y) ->
has_ub (dicho_lb x y P)
x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  intro.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, dicho_lb x y P n <= y
has_ub (dicho_lb x y P)
x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  unfold has_ub.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, dicho_lb x y P n <= y
bound (EUn (dicho_lb x y P))
x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  unfold bound.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, dicho_lb x y P n <= y
exists m : R, is_upper_bound (EUn (dicho_lb x y P)) m
x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  exists y.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, dicho_lb x y P n <= y
is_upper_bound (EUn (dicho_lb x y P)) y
x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  unfold is_upper_bound.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, dicho_lb x y P n <= y
forall x0 : R, EUn (dicho_lb x y P) x0 -> x0 <= y
x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  intros.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, dicho_lb x y P n <= y
x0:R
H1:EUn (dicho_lb x y P) x0
x0 <= y
x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  elim H1; intros.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, dicho_lb x y P n <= y
x0:R
H1:EUn (dicho_lb x y P) x0
x1:nat
H2:x0 = dicho_lb x y P x1
x0 <= y
x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  rewrite H2; apply H0.x, y:R
P:R -> bool
H:x <= y
forall n : nat, dicho_lb x y P n <= y
  apply dicho_lb_maj_y; assumption.
Qed.

Lemma dicho_up_min_x :
  forall (x y:R) (P:R -> bool), x <= y -> forall n:nat, x <= dicho_up x y P n.forall (x y : R) (P : R -> bool),
x <= y -> forall n : nat, x <= dicho_up x y P n
Proof.forall (x y : R) (P : R -> bool),
x <= y -> forall n : nat, x <= dicho_up x y P n
  intros.x, y:R
P:R -> bool
H:x <= y
n:nat
x <= dicho_up x y P n
  induction  n as [| n Hrecn].x, y:R
P:R -> bool
H:x <= y
x <= dicho_up x y P 0
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= dicho_up x y P (S n)
  simpl; assumption.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= dicho_up x y P (S n)
  simpl.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <=
(if
  P
    ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
 then
  (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
 else Dichotomy_ub x y P n)
  case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  unfold Rdiv; apply Rmult_le_reg_l with 2.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
0 < 2
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
2 * x <=
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2)
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  prove_sup0.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
2 * x <=
2 *
((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2)
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  pattern 2 at 1; rewrite Rmult_comm.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
2 * x <=
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 *
2
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ].x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
2 * x <= Dichotomy_lb x y P n + Dichotomy_ub x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  rewrite double; apply Rplus_le_compat.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_lb x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  pattern x at 1; replace x with (Dichotomy_lb x y P 0);
    [ idtac | reflexivity ].x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
Dichotomy_lb x y P 0 <= Dichotomy_lb x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  apply tech9.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
Un_growing (Dichotomy_lb x y P)
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
(0 <= n)%nat
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  assert (H0 := dicho_lb_growing x y P H).x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
H0:Un_growing (dicho_lb x y P)
Un_growing (Dichotomy_lb x y P)
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
(0 <= n)%nat
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  assumption.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
(0 <= n)%nat
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  apply le_O_n.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  assumption.x, y:R
P:R -> bool
H:x <= y
n:nat
Hrecn:x <= dicho_up x y P n
x <= Dichotomy_ub x y P n
  assumption.
Qed.

Lemma dicho_up_min :
  forall (x y:R) (P:R -> bool), x <= y -> has_lb (dicho_up x y P).forall (x y : R) (P : R -> bool),
x <= y -> has_lb (dicho_up x y P)
Proof.forall (x y : R) (P : R -> bool),
x <= y -> has_lb (dicho_up x y P)
  intros.x, y:R
P:R -> bool
H:x <= y
has_lb (dicho_up x y P)
  cut (forall n:nat, x <= dicho_up x y P n).x, y:R
P:R -> bool
H:x <= y
(forall n : nat, x <= dicho_up x y P n) ->
has_lb (dicho_up x y P)
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  intro.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
has_lb (dicho_up x y P)
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  unfold has_lb.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
bound (EUn (opp_seq (dicho_up x y P)))
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  unfold bound.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
exists m : R,
  is_upper_bound (EUn (opp_seq (dicho_up x y P))) m
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  exists (- x).x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
is_upper_bound (EUn (opp_seq (dicho_up x y P))) (- x)
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  unfold is_upper_bound.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
forall x0 : R,
EUn (opp_seq (dicho_up x y P)) x0 -> x0 <= - x
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  intros.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
x0:R
H1:EUn (opp_seq (dicho_up x y P)) x0
x0 <= - x
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  elim H1; intros.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
x0:R
H1:EUn (opp_seq (dicho_up x y P)) x0
x1:nat
H2:x0 = opp_seq (dicho_up x y P) x1
x0 <= - x
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  rewrite H2.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
x0:R
H1:EUn (opp_seq (dicho_up x y P)) x0
x1:nat
H2:x0 = opp_seq (dicho_up x y P) x1
opp_seq (dicho_up x y P) x1 <= - x
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  unfold opp_seq.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
x0:R
H1:EUn (opp_seq (dicho_up x y P)) x0
x1:nat
H2:x0 = opp_seq (dicho_up x y P) x1
- dicho_up x y P x1 <= - x
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  apply Ropp_le_contravar.x, y:R
P:R -> bool
H:x <= y
H0:forall n : nat, x <= dicho_up x y P n
x0:R
H1:EUn (opp_seq (dicho_up x y P)) x0
x1:nat
H2:x0 = opp_seq (dicho_up x y P) x1
x <= dicho_up x y P x1
x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  apply H0.x, y:R
P:R -> bool
H:x <= y
forall n : nat, x <= dicho_up x y P n
  apply dicho_up_min_x; assumption.
Qed.

Lemma dicho_lb_cv :
  forall (x y:R) (P:R -> bool),
    x <= y -> { l:R | Un_cv (dicho_lb x y P) l }.forall (x y : R) (P : R -> bool),
x <= y -> {l : R | Un_cv (dicho_lb x y P) l}
Proof.forall (x y : R) (P : R -> bool),
x <= y -> {l : R | Un_cv (dicho_lb x y P) l}
  intros.x, y:R
P:R -> bool
H:x <= y
{l : R | Un_cv (dicho_lb x y P) l}
  apply growing_cv.x, y:R
P:R -> bool
H:x <= y
Un_growing (dicho_lb x y P)
x, y:R
P:R -> bool
H:x <= y
has_ub (dicho_lb x y P)
  apply dicho_lb_growing; assumption.x, y:R
P:R -> bool
H:x <= y
has_ub (dicho_lb x y P)
  apply dicho_lb_maj; assumption.
Qed.

Lemma dicho_up_cv :
  forall (x y:R) (P:R -> bool),
    x <= y -> { l:R | Un_cv (dicho_up x y P) l }.forall (x y : R) (P : R -> bool),
x <= y -> {l : R | Un_cv (dicho_up x y P) l}
Proof.forall (x y : R) (P : R -> bool),
x <= y -> {l : R | Un_cv (dicho_up x y P) l}
  intros.x, y:R
P:R -> bool
H:x <= y
{l : R | Un_cv (dicho_up x y P) l}
  apply decreasing_cv.x, y:R
P:R -> bool
H:x <= y
Un_decreasing (dicho_up x y P)
x, y:R
P:R -> bool
H:x <= y
has_lb (dicho_up x y P)
  apply dicho_up_decreasing; assumption.x, y:R
P:R -> bool
H:x <= y
has_lb (dicho_up x y P)
  apply dicho_up_min; assumption.
Qed.

Lemma dicho_lb_dicho_up :
  forall (x y:R) (P:R -> bool) (n:nat),
    x <= y -> dicho_up x y P n - dicho_lb x y P n = (y - x) / 2 ^ n.forall (x y : R) (P : R -> bool) (n : nat),
x <= y ->
dicho_up x y P n - dicho_lb x y P n = (y - x) / 2 ^ n
Proof.forall (x y : R) (P : R -> bool) (n : nat),
x <= y ->
dicho_up x y P n - dicho_lb x y P n = (y - x) / 2 ^ n
  intros.x, y:R
P:R -> bool
n:nat
H:x <= y
dicho_up x y P n - dicho_lb x y P n = (y - x) / 2 ^ n
  induction  n as [| n Hrecn].x, y:R
P:R -> bool
H:x <= y
dicho_up x y P 0 - dicho_lb x y P 0 = (y - x) / 2 ^ 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
dicho_up x y P (S n) - dicho_lb x y P (S n) =
(y - x) / 2 ^ S n
  simpl.x, y:R
P:R -> bool
H:x <= y
y - x = (y - x) / 1
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
dicho_up x y P (S n) - dicho_lb x y P (S n) =
(y - x) / 2 ^ S n
  unfold Rdiv; rewrite Rinv_1; ring.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
dicho_up x y P (S n) - dicho_lb x y P (S n) =
(y - x) / 2 ^ S n
  simpl.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(if
  P
    ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
 then
  (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
 else Dichotomy_ub x y P n) -
(if
  P
    ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
 then Dichotomy_lb x y P n
 else
  (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) =
(y - x) / (2 * 2 ^ n)
  case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 -
Dichotomy_lb x y P n = (y - x) / (2 * 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  unfold Rdiv.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 -
Dichotomy_lb x y P n = (y - x) * / (2 * 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  replace
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 - Dichotomy_lb x y P n)
    with ((dicho_up x y P n - dicho_lb x y P n) / 2).x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(y - x) * / (2 * 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 -
Dichotomy_lb x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  unfold Rdiv; rewrite Hrecn.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(y - x) / 2 ^ n * / 2 = (y - x) * / (2 * 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 -
Dichotomy_lb x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  unfold Rdiv.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(y - x) * / 2 ^ n * / 2 = (y - x) * / (2 * 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 -
Dichotomy_lb x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  rewrite Rinv_mult_distr.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(y - x) * / 2 ^ n * / 2 = (y - x) * (/ 2 * / 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 ^ n <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 -
Dichotomy_lb x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  ring.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 ^ n <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 -
Dichotomy_lb x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  discrR.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 ^ n <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 -
Dichotomy_lb x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  apply pow_nonzero; discrR.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 -
Dichotomy_lb x y P n
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  pattern (Dichotomy_lb x y P n) at 2;
    rewrite (double_var (Dichotomy_lb x y P n));
      unfold dicho_up, dicho_lb, Rminus, Rdiv; ring.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
  replace
  (Dichotomy_ub x y P n - (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
    with ((dicho_up x y P n - dicho_lb x y P n) / 2).x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
(y - x) / (2 * 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  unfold Rdiv; rewrite Hrecn.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(y - x) / 2 ^ n * / 2 = (y - x) * / (2 * 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  unfold Rdiv.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(y - x) * / 2 ^ n * / 2 = (y - x) * / (2 * 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  rewrite Rinv_mult_distr.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(y - x) * / 2 ^ n * / 2 = (y - x) * (/ 2 * / 2 ^ n)
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 ^ n <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  ring.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 ^ n <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  discrR.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
2 ^ n <> 0
x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  apply pow_nonzero; discrR.x, y:R
P:R -> bool
n:nat
H:x <= y
Hrecn:dicho_up x y P n - dicho_lb x y P n =
(y - x) / 2 ^ n
(dicho_up x y P n - dicho_lb x y P n) / 2 =
Dichotomy_ub x y P n -
(Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
  pattern (Dichotomy_ub x y P n) at 1;
    rewrite (double_var (Dichotomy_ub x y P n));
      unfold dicho_up, dicho_lb, Rminus, Rdiv; ring.
Qed.

Definition pow_2_n (n:nat) := 2 ^ n.

Lemma pow_2_n_neq_R0 : forall n:nat, pow_2_n n <> 0.forall n : nat, pow_2_n n <> 0
Proof.forall n : nat, pow_2_n n <> 0
  intro.n:nat
pow_2_n n <> 0
  unfold pow_2_n.n:nat
2 ^ n <> 0
  apply pow_nonzero.n:nat
2 <> 0
  discrR.
Qed.

Lemma pow_2_n_growing : Un_growing pow_2_n.Un_growing pow_2_n
Proof.Un_growing pow_2_n
  unfold Un_growing.forall n : nat, pow_2_n n <= pow_2_n (S n)
  intro.n:nat
pow_2_n n <= pow_2_n (S n)
  replace (S n) with (n + 1)%nat;
  [ unfold pow_2_n; rewrite pow_add | ring ].n:nat
2 ^ n <= 2 ^ n * 2 ^ 1
  pattern (2 ^ n) at 1; rewrite <- Rmult_1_r.n:nat
2 ^ n * 1 <= 2 ^ n * 2 ^ 1
  apply Rmult_le_compat_l.n:nat
0 <= 2 ^ n
n:nat
1 <= 2 ^ 1
  left; apply pow_lt; prove_sup0.n:nat
1 <= 2 ^ 1
  simpl.n:nat
1 <= 2 * 1
  rewrite Rmult_1_r.n:nat
1 <= 2
  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
    apply Rlt_0_1.
Qed.

Lemma pow_2_n_infty : cv_infty pow_2_n.cv_infty pow_2_n
Proof.cv_infty pow_2_n
  cut (forall N:nat, INR N <= 2 ^ N).(forall N : nat, INR N <= 2 ^ N) -> cv_infty pow_2_n
forall N : nat, INR N <= 2 ^ N
  intros.H:forall N : nat, INR N <= 2 ^ N
cv_infty pow_2_n
forall N : nat, INR N <= 2 ^ N
  unfold cv_infty.H:forall N : nat, INR N <= 2 ^ N
forall M : R,
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  intro.H:forall N : nat, INR N <= 2 ^ N
M:R
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  destruct (total_order_T 0 M) as [[Hlt|<-]|Hgt].H:forall N : nat, INR N <= 2 ^ N
M:R
Hlt:0 < M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  set (N := up M).H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
exists N0 : nat,
  forall n : nat, (N0 <= n)%nat -> M < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  cut (0 <= N)%Z.H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z ->
exists N0 : nat,
  forall n : nat, (N0 <= n)%nat -> M < pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  intro.H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
exists N0 : nat,
  forall n : nat, (N0 <= n)%nat -> M < pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  elim (IZN N H0); intros N0 H1.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
exists N1 : nat,
  forall n : nat, (N1 <= n)%nat -> M < pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  exists N0.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
forall n : nat, (N0 <= n)%nat -> M < pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  intros.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
M < pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  apply Rlt_le_trans with (INR N0).H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
M < INR N0
H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
INR N0 <= pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  rewrite INR_IZR_INZ.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
M < IZR (Z.of_nat N0)
H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
INR N0 <= pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  rewrite <- H1.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
M < IZR N
H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
INR N0 <= pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  unfold N.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
M < IZR (up M)
H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
INR N0 <= pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  assert (H3 := archimed M).H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
H3:IZR (up M) > M /\ IZR (up M) - M <= 1
M < IZR (up M)
H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
INR N0 <= pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  elim H3; intros; assumption.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
INR N0 <= pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  apply Rle_trans with (pow_2_n N0).H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
INR N0 <= pow_2_n N0
H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
pow_2_n N0 <= pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  unfold pow_2_n; apply H.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
pow_2_n N0 <= pow_2_n n
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  apply Rge_le.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
pow_2_n n >= pow_2_n N0
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  apply growing_prop.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
Un_growing pow_2_n
H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
(n >= N0)%nat
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  apply pow_2_n_growing.H:forall N1 : nat, INR N1 <= 2 ^ N1
M:R
Hlt:0 < M
N:=up M:Z
H0:(0 <= N)%Z
N0:nat
H1:N = Z.of_nat N0
n:nat
H2:(N0 <= n)%nat
(n >= N0)%nat
H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  assumption.H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
(0 <= N)%Z
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  apply le_IZR.H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
0 <= IZR N
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  unfold N.H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
0 <= IZR (up M)
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  simpl.H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
0 <= IZR (up M)
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  assert (H0 := archimed M); elim H0; intros.H:forall N0 : nat, INR N0 <= 2 ^ N0
M:R
Hlt:0 < M
N:=up M:Z
H0:IZR (up M) > M /\ IZR (up M) - M <= 1
H1:IZR (up M) > M
H2:IZR (up M) - M <= 1
0 <= IZR (up M)
H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  left; apply Rlt_trans with M; assumption.H:forall N : nat, INR N <= 2 ^ N
exists N : nat,
  forall n : nat, (N <= n)%nat -> 0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  exists 0%nat; intros.H:forall N : nat, INR N <= 2 ^ N
n:nat
H0:(0 <= n)%nat
0 < pow_2_n n
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  unfold pow_2_n; apply pow_lt; prove_sup0.H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
exists N : nat,
  forall n : nat, (N <= n)%nat -> M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  exists 0%nat; intros.H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
n:nat
H0:(0 <= n)%nat
M < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  apply Rlt_trans with 0.H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
n:nat
H0:(0 <= n)%nat
M < 0
H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
n:nat
H0:(0 <= n)%nat
0 < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  assumption.H:forall N : nat, INR N <= 2 ^ N
M:R
Hgt:0 > M
n:nat
H0:(0 <= n)%nat
0 < pow_2_n n
forall N : nat, INR N <= 2 ^ N
  unfold pow_2_n; apply pow_lt; prove_sup0.forall N : nat, INR N <= 2 ^ N
  simple induction N.N:nat
INR 0 <= 2 ^ 0
N:nat
forall n : nat, INR n <= 2 ^ n -> INR (S n) <= 2 ^ S n
  simpl.N:nat
0 <= 1
N:nat
forall n : nat, INR n <= 2 ^ n -> INR (S n) <= 2 ^ S n
  left; apply Rlt_0_1.N:nat
forall n : nat, INR n <= 2 ^ n -> INR (S n) <= 2 ^ S n
  intros.N, n:nat
H:INR n <= 2 ^ n
INR (S n) <= 2 ^ S n
  pattern (S n) at 2; replace (S n) with (n + 1)%nat; [ idtac | ring ].N, n:nat
H:INR n <= 2 ^ n
INR (S n) <= 2 ^ (n + 1)
  rewrite S_INR; rewrite pow_add.N, n:nat
H:INR n <= 2 ^ n
INR n + 1 <= 2 ^ n * 2 ^ 1
  simpl.N, n:nat
H:INR n <= 2 ^ n
INR n + 1 <= 2 ^ n * (2 * 1)
  rewrite Rmult_1_r.N, n:nat
H:INR n <= 2 ^ n
INR n + 1 <= 2 ^ n * 2
  apply Rle_trans with (2 ^ n).N, n:nat
H:INR n <= 2 ^ n
INR n + 1 <= 2 ^ n
N, n:nat
H:INR n <= 2 ^ n
2 ^ n <= 2 ^ n * 2
  rewrite <- (Rplus_comm 1).N, n:nat
H:INR n <= 2 ^ n
1 + INR n <= 2 ^ n
N, n:nat
H:INR n <= 2 ^ n
2 ^ n <= 2 ^ n * 2
  rewrite <- (Rmult_1_r (INR n)).N, n:nat
H:INR n <= 2 ^ n
1 + INR n * 1 <= 2 ^ n
N, n:nat
H:INR n <= 2 ^ n
2 ^ n <= 2 ^ n * 2
  apply (poly n 1).N, n:nat
H:INR n <= 2 ^ n
0 < 1
N, n:nat
H:INR n <= 2 ^ n
2 ^ n <= 2 ^ n * 2
  apply Rlt_0_1.N, n:nat
H:INR n <= 2 ^ n
2 ^ n <= 2 ^ n * 2
  pattern (2 ^ n) at 1; rewrite <- Rplus_0_r.N, n:nat
H:INR n <= 2 ^ n
2 ^ n + 0 <= 2 ^ n * 2
  rewrite <- (Rmult_comm 2).N, n:nat
H:INR n <= 2 ^ n
2 ^ n + 0 <= 2 * 2 ^ n
  rewrite double.N, n:nat
H:INR n <= 2 ^ n
2 ^ n + 0 <= 2 ^ n + 2 ^ n
  apply Rplus_le_compat_l.N, n:nat
H:INR n <= 2 ^ n
0 <= 2 ^ n
  left; apply pow_lt; prove_sup0.
Qed.

Lemma cv_dicho :
  forall (x y l1 l2:R) (P:R -> bool),
    x <= y ->
    Un_cv (dicho_lb x y P) l1 -> Un_cv (dicho_up x y P) l2 -> l1 = l2.forall (x y l1 l2 : R) (P : R -> bool),
x <= y ->
Un_cv (dicho_lb x y P) l1 ->
Un_cv (dicho_up x y P) l2 -> l1 = l2
Proof.forall (x y l1 l2 : R) (P : R -> bool),
x <= y ->
Un_cv (dicho_lb x y P) l1 ->
Un_cv (dicho_up x y P) l2 -> l1 = l2
  intros.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
l1 = l2
  assert (H2 := CV_minus _ _ _ _ H0 H1).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
l1 = l2
  cut (Un_cv (fun i:nat => dicho_lb x y P i - dicho_up x y P i) 0).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
Un_cv
  (fun i : nat => dicho_lb x y P i - dicho_up x y P i)
  0 -> l1 = l2
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
Un_cv
  (fun i : nat => dicho_lb x y P i - dicho_up x y P i)
  0
  intro.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
H3:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) 0
l1 = l2
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
Un_cv
  (fun i : nat => dicho_lb x y P i - dicho_up x y P i)
  0
  assert (H4 := UL_sequence _ _ _ H2 H3).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
H3:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) 0
H4:l1 - l2 = 0
l1 = l2
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
Un_cv
  (fun i : nat => dicho_lb x y P i - dicho_up x y P i)
  0
  symmetry ; apply Rminus_diag_uniq_sym; assumption.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
Un_cv
  (fun i : nat => dicho_lb x y P i - dicho_up x y P i)
  0
  unfold Un_cv; unfold R_dist.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  intros.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  assert (H4 := cv_infty_cv_R0 pow_2_n pow_2_n_neq_R0 pow_2_n_infty).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  destruct (total_order_T x y) as [[ Hlt | -> ]|Hgt].x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hlt:x < y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  unfold Un_cv in H4; unfold R_dist in H4.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  cut (0 < y - x).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  intro Hyp.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  cut (0 < eps / (y - x)).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x) ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  intro.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  elim (H4 (eps / (y - x)) H5); intros N H6.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n : nat,
(n >= N)%nat ->
Rabs (/ pow_2_n n - 0) < eps / (y - x)
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  exists N; intros.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  replace (dicho_lb x y P n - dicho_up x y P n - 0) with
  (dicho_lb x y P n - dicho_up x y P n); [ idtac | ring ].x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
Rabs (dicho_lb x y P n - dicho_up x y P n) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite <- Rabs_Ropp.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
Rabs (- (dicho_lb x y P n - dicho_up x y P n)) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite Ropp_minus_distr'.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
Rabs (dicho_up x y P n - dicho_lb x y P n) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite dicho_lb_dicho_up.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
Rabs ((y - x) / 2 ^ n) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  unfold Rdiv; rewrite Rabs_mult.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
Rabs (y - x) * Rabs (/ 2 ^ n) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite (Rabs_right (y - x)).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
(y - x) * Rabs (/ 2 ^ n) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x >= 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  apply Rmult_lt_reg_l with (/ (y - x)).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
0 < / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
/ (y - x) * ((y - x) * Rabs (/ 2 ^ n)) <
/ (y - x) * eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x >= 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  apply Rinv_0_lt_compat; assumption.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
/ (y - x) * ((y - x) * Rabs (/ 2 ^ n)) <
/ (y - x) * eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x >= 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
1 * Rabs (/ 2 ^ n) < / (y - x) * eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x <> 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x >= 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite Rmult_1_l.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
Rabs (/ 2 ^ n) < / (y - x) * eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x <> 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x >= 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  replace (/ 2 ^ n) with (/ 2 ^ n - 0);
  [ unfold pow_2_n, Rdiv in H6; rewrite <- (Rmult_comm eps); apply H6;
    assumption
    | ring ].x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x <> 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x >= 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  red; intro; rewrite H8 in Hyp; elim (Rlt_irrefl _ Hyp).x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
y - x >= 0
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  apply Rle_ge.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
0 <= y - x
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  apply Rplus_le_reg_l with x; rewrite Rplus_0_r.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= x + (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  replace (x + (y - x)) with y; [ assumption | ring ].x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (/ pow_2_n n0 - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
H5:0 < eps / (y - x)
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (/ pow_2_n n0 - 0) < eps / (y - x)
n:nat
H7:(n >= N)%nat
x <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  assumption.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
Hyp:0 < y - x
0 < eps / (y - x)
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  unfold Rdiv; apply Rmult_lt_0_compat;
    [ assumption | apply Rinv_0_lt_compat; assumption ].x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
0 < y - x
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  apply Rplus_lt_reg_l with x; rewrite Rplus_0_r.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (/ pow_2_n n - 0) < eps0
Hlt:x < y
x < x + (y - x)
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  replace (x + (y - x)) with y; [ assumption | ring ].y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  exists 0%nat; intros.y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n0 : nat => / pow_2_n n0) 0
n:nat
H5:(n >= 0)%nat
Rabs (dicho_lb y y P n - dicho_up y y P n - 0) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  replace (dicho_lb y y P n - dicho_up y y P n - 0) with
  (dicho_lb y y P n - dicho_up y y P n); [ idtac | ring ].y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n0 : nat => / pow_2_n n0) 0
n:nat
H5:(n >= 0)%nat
Rabs (dicho_lb y y P n - dicho_up y y P n) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite <- Rabs_Ropp.y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n0 : nat => / pow_2_n n0) 0
n:nat
H5:(n >= 0)%nat
Rabs (- (dicho_lb y y P n - dicho_up y y P n)) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite Ropp_minus_distr'.y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n0 : nat => / pow_2_n n0) 0
n:nat
H5:(n >= 0)%nat
Rabs (dicho_up y y P n - dicho_lb y y P n) < eps
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  rewrite dicho_lb_dicho_up.y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n0 : nat => / pow_2_n n0) 0
n:nat
H5:(n >= 0)%nat
Rabs ((y - y) / 2 ^ n) < eps
y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n0 : nat => / pow_2_n n0) 0
n:nat
H5:(n >= 0)%nat
y <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  unfold Rminus, Rdiv; rewrite Rplus_opp_r; rewrite Rmult_0_l;
    rewrite Rabs_R0; assumption.y, l1, l2:R
P:R -> bool
H2:Un_cv
  (fun i : nat =>
   dicho_lb y y P i - dicho_up y y P i) (l1 - l2)
H1:Un_cv (dicho_up y y P) l2
H0:Un_cv (dicho_lb y y P) l1
H:y <= y
eps:R
H3:eps > 0
H4:Un_cv (fun n0 : nat => / pow_2_n n0) 0
n:nat
H5:(n >= 0)%nat
y <= y
x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  assumption.x, y, l1, l2:R
P:R -> bool
H:x <= y
H0:Un_cv (dicho_lb x y P) l1
H1:Un_cv (dicho_up x y P) l2
H2:Un_cv
  (fun i : nat =>
   dicho_lb x y P i - dicho_up x y P i) (l1 - l2)
eps:R
H3:eps > 0
H4:Un_cv (fun n : nat => / pow_2_n n) 0
Hgt:x > y
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (dicho_lb x y P n - dicho_up x y P n - 0) < eps
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H Hgt)).
Qed.

Definition cond_positivity (x:R) : bool :=
  match Rle_dec 0 x with
    | left _ => true
    | right _ => false
  end.

Sequential characterisation of continuity

Lemma continuity_seq :
  forall (f:R -> R) (Un:nat -> R) (l:R),
    continuity_pt f l -> Un_cv Un l -> Un_cv (fun i:nat => f (Un i)) (f l).forall (f : R -> R) (Un : nat -> R) (l : R),
continuity_pt f l ->
Un_cv Un l -> Un_cv (fun i : nat => f (Un i)) (f l)
Proof.forall (f : R -> R) (Un : nat -> R) (l : R),
continuity_pt f l ->
Un_cv Un l -> Un_cv (fun i : nat => f (Un i)) (f l)
  unfold continuity_pt, Un_cv; unfold continue_in.forall (f : R -> R) (Un : nat -> R) (l : R),
limit1_in f (D_x no_cond l) (f l) l ->
(forall eps : R,
 eps > 0 ->
 exists N : nat,
   forall n : nat,
   (n >= N)%nat -> R_dist (Un n) l < eps) ->
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (f (Un n)) (f l) < eps
  unfold limit1_in.forall (f : R -> R) (Un : nat -> R) (l : R),
limit_in R_met R_met f (D_x no_cond l) l (f l) ->
(forall eps : R,
 eps > 0 ->
 exists N : nat,
   forall n : nat,
   (n >= N)%nat -> R_dist (Un n) l < eps) ->
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (f (Un n)) (f l) < eps
  unfold limit_in.forall (f : R -> R) (Un : nat -> R) (l : R),
(forall eps : R,
 eps > 0 ->
 exists alp : R,
   alp > 0 /\
   (forall x : Base R_met,
    D_x no_cond l x /\ dist R_met x l < alp ->
    dist R_met (f x) (f l) < eps)) ->
(forall eps : R,
 eps > 0 ->
 exists N : nat,
   forall n : nat,
   (n >= N)%nat -> R_dist (Un n) l < eps) ->
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (f (Un n)) (f l) < eps
  unfold dist.forall (f : R -> R) (Un : nat -> R) (l : R),
(forall eps : R,
 eps > 0 ->
 exists alp : R,
   alp > 0 /\
   (forall x : Base R_met,
    D_x no_cond l x /\
    (let
       (Base, dist, _, _, _, _) as m
        return (Base m -> Base m -> R) := R_met in
     dist) x l < alp ->
    (let
       (Base, dist, _, _, _, _) as m
        return (Base m -> Base m -> R) := R_met in
     dist) (f x) (f l) < eps)) ->
(forall eps : R,
 eps > 0 ->
 exists N : nat,
   forall n : nat,
   (n >= N)%nat -> R_dist (Un n) l < eps) ->
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (f (Un n)) (f l) < eps
  simpl.forall (f : R -> R) (Un : nat -> R) (l : R),
(forall eps : R,
 eps > 0 ->
 exists alp : R,
   alp > 0 /\
   (forall x : R,
    D_x no_cond l x /\ R_dist x l < alp ->
    R_dist (f x) (f l) < eps)) ->
(forall eps : R,
 eps > 0 ->
 exists N : nat,
   forall n : nat,
   (n >= N)%nat -> R_dist (Un n) l < eps) ->
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (f (Un n)) (f l) < eps
  unfold R_dist.forall (f : R -> R) (Un : nat -> R) (l : R),
(forall eps : R,
 eps > 0 ->
 exists alp : R,
   alp > 0 /\
   (forall x : R,
    D_x no_cond l x /\ Rabs (x - l) < alp ->
    Rabs (f x - f l) < eps)) ->
(forall eps : R,
 eps > 0 ->
 exists N : nat,
   forall n : nat,
   (n >= N)%nat -> Rabs (Un n - l) < eps) ->
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Un n) - f l) < eps
  intros.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Un n - l) < eps0
eps:R
H1:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Un n) - f l) < eps
  elim (H eps H1); intros alp H2.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Un n - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Un n) - f l) < eps
  elim H2; intros.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Un n - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Un n) - f l) < eps
  elim (H0 alp H3); intros N H5.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> Rabs (Un n - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n : nat,
(n >= N)%nat -> Rabs (Un n - l) < alp
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> Rabs (f (Un n) - f l) < eps
  exists N; intros.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
Rabs (f (Un n) - f l) < eps
  case (Req_dec (Un n) l); intro.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n = l
Rabs (f (Un n) - f l) < eps
f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
Rabs (f (Un n) - f l) < eps
  rewrite H7; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
    assumption.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
Rabs (f (Un n) - f l) < eps
  apply H4.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
D_x no_cond l (Un n) /\ Rabs (Un n - l) < alp
  split.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
D_x no_cond l (Un n)
f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
Rabs (Un n - l) < alp
  unfold D_x, no_cond.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
True /\ l <> Un n
f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
Rabs (Un n - l) < alp
  split.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
True
f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
l <> Un n
f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
Rabs (Un n - l) < alp
  trivial.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
l <> Un n
f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
Rabs (Un n - l) < alp
  apply (not_eq_sym (A:=R)); assumption.f:R -> R
Un:nat -> R
l:R
H:forall eps0 : R,
eps0 > 0 ->
exists alp0 : R,
  alp0 > 0 /\
  (forall x : R,
   D_x no_cond l x /\ Rabs (x - l) < alp0 ->
   Rabs (f x - f l) < eps0)
H0:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat -> Rabs (Un n0 - l) < eps0
eps:R
H1:eps > 0
alp:R
H2:alp > 0 /\
(forall x : R,
 D_x no_cond l x /\ Rabs (x - l) < alp ->
 Rabs (f x - f l) < eps)
H3:alp > 0
H4:forall x : R,
D_x no_cond l x /\ Rabs (x - l) < alp ->
Rabs (f x - f l) < eps
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat -> Rabs (Un n0 - l) < alp
n:nat
H6:(n >= N)%nat
H7:Un n <> l
Rabs (Un n - l) < alp
  apply H5; assumption.
Qed.

Lemma dicho_lb_car :
  forall (x y:R) (P:R -> bool) (n:nat),
    P x = false -> P (dicho_lb x y P n) = false.forall (x y : R) (P : R -> bool) (n : nat),
P x = false -> P (dicho_lb x y P n) = false
Proof.forall (x y : R) (P : R -> bool) (n : nat),
P x = false -> P (dicho_lb x y P n) = false
  intros.x, y:R
P:R -> bool
n:nat
H:P x = false
P (dicho_lb x y P n) = false
  induction n as [| n Hrecn].x, y:R
P:R -> bool
H:P x = false
P (dicho_lb x y P 0) = false
x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
P (dicho_lb x y P (S n)) = false
  -x, y:R
P:R -> bool
H:P x = false
P (dicho_lb x y P 0) = false assumption.
  -x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
P (dicho_lb x y P (S n)) = false simpl.x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then Dichotomy_lb x y P n
   else
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) =
false
    destruct
        (sumbool_of_bool (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2))) as [Heq|Heq].x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = true
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then Dichotomy_lb x y P n
   else
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) =
false
x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = false
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then Dichotomy_lb x y P n
   else
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) =
false
    +x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = true
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then Dichotomy_lb x y P n
   else
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) =
false rewrite Heq.x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = true
P (Dichotomy_lb x y P n) = false
      unfold dicho_lb in Hrecn; assumption.
    +x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = false
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then Dichotomy_lb x y P n
   else
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) =
false rewrite Heq.x, y:R
P:R -> bool
n:nat
H:P x = false
Hrecn:P (dicho_lb x y P n) = false
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = false
P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) =
false
      assumption.
Qed.

Lemma dicho_up_car :
  forall (x y:R) (P:R -> bool) (n:nat),
    P y = true -> P (dicho_up x y P n) = true.forall (x y : R) (P : R -> bool) (n : nat),
P y = true -> P (dicho_up x y P n) = true
Proof.forall (x y : R) (P : R -> bool) (n : nat),
P y = true -> P (dicho_up x y P n) = true
  intros.x, y:R
P:R -> bool
n:nat
H:P y = true
P (dicho_up x y P n) = true
  induction n as [| n Hrecn].x, y:R
P:R -> bool
H:P y = true
P (dicho_up x y P 0) = true
x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
P (dicho_up x y P (S n)) = true
  -x, y:R
P:R -> bool
H:P y = true
P (dicho_up x y P 0) = true assumption.
  -x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
P (dicho_up x y P (S n)) = true simpl.x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
   else Dichotomy_ub x y P n) = true
    destruct
      (sumbool_of_bool (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2))) as [Heq|Heq].x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = true
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
   else Dichotomy_ub x y P n) = true
x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = false
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
   else Dichotomy_ub x y P n) = true
    +x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = true
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
   else Dichotomy_ub x y P n) = true rewrite Heq.x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = true
P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) =
true
      unfold dicho_lb in Hrecn; assumption.
    +x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = false
P
  (if
    P
      ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
       2)
   then
    (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2
   else Dichotomy_ub x y P n) = true rewrite  Heq.x, y:R
P:R -> bool
n:nat
H:P y = true
Hrecn:P (dicho_up x y P n) = true
Heq:P
  ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) /
   2) = false
P (Dichotomy_ub x y P n) = true
      assumption.
Qed.

(* A general purpose corollary. *)
Lemma cv_pow_half : forall a, Un_cv (fun n => a/2^n) 0.forall a : R, Un_cv (fun n : nat => a / 2 ^ n) 0
intros a; unfold Rdiv; replace 0 with (a * 0) by ring.a:R
Un_cv (fun n : nat => a * / 2 ^ n) (a * 0)
apply CV_mult.a:R
Un_cv (fun _ : nat => a) a
a:R
Un_cv (fun i : nat => / 2 ^ i) 0
 intros eps ep; exists 0%nat; rewrite R_dist_eq; intros n _; assumption.a:R
Un_cv (fun i : nat => / 2 ^ i) 0
exact (cv_infty_cv_R0 pow_2_n pow_2_n_neq_R0 pow_2_n_infty).
Qed.

Intermediate Value Theorem

Lemma IVT :
  forall (f:R -> R) (x y:R),
    continuity f ->
    x < y -> f x < 0 -> 0 < f y -> { z:R | x <= z <= y /\ f z = 0 }.forall (f : R -> R) (x y : R),
continuity f ->
x < y ->
f x < 0 -> 0 < f y -> {z : R | x <= z <= y /\ f z = 0}
Proof.forall (f : R -> R) (x y : R),
continuity f ->
x < y ->
f x < 0 -> 0 < f y -> {z : R | x <= z <= y /\ f z = 0}
  intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
{z : R | x <= z <= y /\ f z = 0}
  assert (x <= y) by (left; assumption).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
{z : R | x <= z <= y /\ f z = 0}
  destruct (dicho_lb_cv x y (fun z:R => cond_positivity (f z)) H3) as (x1,p0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x1
{z : R | x <= z <= y /\ f z = 0}
  destruct (dicho_up_cv x y (fun z:R => cond_positivity (f z)) H3) as (x0,p).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x1
x0:R
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
{z : R | x <= z <= y /\ f z = 0}
  assert (H4 := cv_dicho _ _ _ _ _ H3 p0 p).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x1
x0:R
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
{z : R | x <= z <= y /\ f z = 0}
  rewrite H4 in p0.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
{z : R | x <= z <= y /\ f z = 0}
  exists x0.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x <= x0 <= y /\ f x0 = 0
  split.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x <= x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  split.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x <= x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  apply Rle_trans with (dicho_lb x y (fun z:R => cond_positivity (f z)) 0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x <=
dicho_lb x y (fun z : R => cond_positivity (f z)) 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
dicho_lb x y (fun z : R => cond_positivity (f z)) 0 <=
x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  simpl.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x <= x
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
dicho_lb x y (fun z : R => cond_positivity (f z)) 0 <=
x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  right; reflexivity.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
dicho_lb x y (fun z : R => cond_positivity (f z)) 0 <=
x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  apply growing_ineq.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Un_growing
  (dicho_lb x y (fun z : R => cond_positivity (f z)))
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Un_cv
  (dicho_lb x y (fun z : R => cond_positivity (f z)))
  x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  apply dicho_lb_growing; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Un_cv
  (dicho_lb x y (fun z : R => cond_positivity (f z)))
  x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  apply Rle_trans with (dicho_up x y (fun z:R => cond_positivity (f z)) 0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
x0 <=
dicho_up x y (fun z : R => cond_positivity (f z)) 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
dicho_up x y (fun z : R => cond_positivity (f z)) 0 <=
y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  apply decreasing_ineq.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Un_decreasing
  (dicho_up x y (fun z : R => cond_positivity (f z)))
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Un_cv
  (dicho_up x y (fun z : R => cond_positivity (f z)))
  x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
dicho_up x y (fun z : R => cond_positivity (f z)) 0 <=
y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  apply dicho_up_decreasing; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Un_cv
  (dicho_up x y (fun z : R => cond_positivity (f z)))
  x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
dicho_up x y (fun z : R => cond_positivity (f z)) 0 <=
y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
dicho_up x y (fun z : R => cond_positivity (f z)) 0 <=
y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  right; reflexivity.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
f x0 = 0
  set (Vn := fun n:nat => dicho_lb x y (fun z:R => cond_positivity (f z)) n).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
f x0 = 0
  set (Wn := fun n:nat => dicho_up x y (fun z:R => cond_positivity (f z)) n).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
f x0 = 0
  cut ((forall n:nat, f (Vn n) <= 0) -> f x0 <= 0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
((forall n : nat, f (Vn n) <= 0) -> f x0 <= 0) ->
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut ((forall n:nat, 0 <= f (Wn n)) -> 0 <= f x0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
((forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0) ->
((forall n : nat, f (Vn n) <= 0) -> f x0 <= 0) ->
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut (forall n:nat, f (Vn n) <= 0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
(forall n : nat, f (Vn n) <= 0) -> f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut (forall n:nat, 0 <= f (Wn n)).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
(forall n : nat, 0 <= f (Wn n)) ->
(forall n : nat, f (Vn n) <= 0) -> f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, 0 <= f (Wn n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
H7:forall n : nat, 0 <= f (Wn n)
H8:forall n : nat, f (Vn n) <= 0
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, 0 <= f (Wn n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assert (H9 := H6 H8).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
H7:forall n : nat, 0 <= f (Wn n)
H8:forall n : nat, f (Vn n) <= 0
H9:f x0 <= 0
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, 0 <= f (Wn n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assert (H10 := H5 H7).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
H7:forall n : nat, 0 <= f (Wn n)
H8:forall n : nat, f (Vn n) <= 0
H9:f x0 <= 0
H10:0 <= f x0
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, 0 <= f (Wn n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply Rle_antisym; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, 0 <= f (Wn n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
0 <= f (Wn n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  unfold Wn.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
0 <=
f
  (dicho_up x y (fun z : R => cond_positivity (f z)) n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut (forall z:R, cond_positivity z = true <-> 0 <= z).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
(forall z : R, cond_positivity z = true <-> 0 <= z) ->
0 <=
f
  (dicho_up x y (fun z : R => cond_positivity (f z)) n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
H7:forall z : R, cond_positivity z = true <-> 0 <= z
0 <=
f
  (dicho_up x y (fun z : R => cond_positivity (f z)) n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assert (H8 := dicho_up_car x y (fun z:R => cond_positivity (f z)) n).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
H7:forall z : R, cond_positivity z = true <-> 0 <= z
H8:(fun z : R => cond_positivity (f z)) y = true ->
(fun z : R => cond_positivity (f z))
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n) =
true
0 <=
f
  (dicho_up x y (fun z : R => cond_positivity (f z)) n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  elim (H7 (f (dicho_up x y (fun z:R => cond_positivity (f z)) n))); intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
H7:forall z : R, cond_positivity z = true <-> 0 <= z
H8:(fun z : R => cond_positivity (f z)) y = true ->
(fun z : R => cond_positivity (f z))
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n) =
true
H9:cond_positivity
  (f
     (dicho_up x y
        (fun z : R => cond_positivity (f z)) n)) =
true ->
0 <=
f
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n)
H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) ->
cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) =
true
0 <=
f
  (dicho_up x y (fun z : R => cond_positivity (f z)) n)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply H9.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
H7:forall z : R, cond_positivity z = true <-> 0 <= z
H8:(fun z : R => cond_positivity (f z)) y = true ->
(fun z : R => cond_positivity (f z))
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n) =
true
H9:cond_positivity
  (f
     (dicho_up x y
        (fun z : R => cond_positivity (f z)) n)) =
true ->
0 <=
f
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n)
H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) ->
cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) =
true
cond_positivity
  (f
     (dicho_up x y
        (fun z : R => cond_positivity (f z)) n)) =
true
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply H8.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
H7:forall z : R, cond_positivity z = true <-> 0 <= z
H8:(fun z : R => cond_positivity (f z)) y = true ->
(fun z : R => cond_positivity (f z))
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n) =
true
H9:cond_positivity
  (f
     (dicho_up x y
        (fun z : R => cond_positivity (f z)) n)) =
true ->
0 <=
f
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n)
H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) ->
cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) =
true
cond_positivity (f y) = true
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  elim (H7 (f y)); intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
H7:forall z : R, cond_positivity z = true <-> 0 <= z
H8:(fun z : R => cond_positivity (f z)) y = true ->
(fun z : R => cond_positivity (f z))
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n) =
true
H9:cond_positivity
  (f
     (dicho_up x y
        (fun z : R => cond_positivity (f z)) n)) =
true ->
0 <=
f
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n)
H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) ->
cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) =
true
H11:cond_positivity (f y) = true -> 0 <= f y
H12:0 <= f y -> cond_positivity (f y) = true
cond_positivity (f y) = true
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply H12.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
H7:forall z : R, cond_positivity z = true <-> 0 <= z
H8:(fun z : R => cond_positivity (f z)) y = true ->
(fun z : R => cond_positivity (f z))
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n) =
true
H9:cond_positivity
  (f
     (dicho_up x y
        (fun z : R => cond_positivity (f z)) n)) =
true ->
0 <=
f
  (dicho_up x y
     (fun z : R => cond_positivity (f z)) n)
H10:0 <= f (dicho_up x y (fun z : R => cond_positivity (f z)) n) ->
cond_positivity (f (dicho_up x y (fun z : R => cond_positivity (f z)) n)) =
true
H11:cond_positivity (f y) = true -> 0 <= f y
H12:0 <= f y -> cond_positivity (f y) = true
0 <= f y
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  left; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
forall z : R, cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
cond_positivity z = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  unfold cond_positivity.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
(if Rle_dec 0 z then true else false) = true <->
0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  case (Rle_dec 0 z) as [Hle|Hnle].f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hle:0 <= z
true = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hnle:~ 0 <= z
false = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  split.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hle:0 <= z
true = true -> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hle:0 <= z
0 <= z -> true = true
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hnle:~ 0 <= z
false = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hle:0 <= z
0 <= z -> true = true
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hnle:~ 0 <= z
false = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro; reflexivity.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hnle:~ 0 <= z
false = true <-> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  split.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hnle:~ 0 <= z
false = true -> 0 <= z
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hnle:~ 0 <= z
0 <= z -> false = true
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro feqt;discriminate feqt.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hnle:~ 0 <= z
0 <= z -> false = true
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
n:nat
z:R
Hnle:~ 0 <= z
H7:0 <= z
false = true
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  contradiction.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat, f (Vn n) <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  unfold Vn.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall n : nat,
f
  (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <=
0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut (forall z:R, cond_positivity z = false <-> z < 0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
(forall z : R, cond_positivity z = false <-> z < 0) ->
forall n : nat,
f
  (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <=
0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
H7:forall z : R, cond_positivity z = false <-> z < 0
n:nat
f
  (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <=
0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assert (H8 := dicho_lb_car x y (fun z:R => cond_positivity (f z)) n).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
H7:forall z : R, cond_positivity z = false <-> z < 0
n:nat
H8:(fun z : R => cond_positivity (f z)) x = false ->
(fun z : R => cond_positivity (f z))
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) =
false
f
  (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <=
0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  left.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
H7:forall z : R, cond_positivity z = false <-> z < 0
n:nat
H8:(fun z : R => cond_positivity (f z)) x = false ->
(fun z : R => cond_positivity (f z))
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) =
false
f
  (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <
0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  elim (H7 (f (dicho_lb x y (fun z:R => cond_positivity (f z)) n))); intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
H7:forall z : R, cond_positivity z = false <-> z < 0
n:nat
H8:(fun z : R => cond_positivity (f z)) x = false ->
(fun z : R => cond_positivity (f z))
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) =
false
H9:cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false ->
f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0
H10:f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0 ->
cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false
f
  (dicho_lb x y (fun z : R => cond_positivity (f z)) n) <
0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply H9.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
H7:forall z : R, cond_positivity z = false <-> z < 0
n:nat
H8:(fun z : R => cond_positivity (f z)) x = false ->
(fun z : R => cond_positivity (f z))
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) =
false
H9:cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false ->
f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0
H10:f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0 ->
cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false
cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply H8.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
H7:forall z : R, cond_positivity z = false <-> z < 0
n:nat
H8:(fun z : R => cond_positivity (f z)) x = false ->
(fun z : R => cond_positivity (f z))
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) =
false
H9:cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false ->
f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0
H10:f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0 ->
cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false
cond_positivity (f x) = false
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  elim (H7 (f x)); intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
H7:forall z : R, cond_positivity z = false <-> z < 0
n:nat
H8:(fun z : R => cond_positivity (f z)) x = false ->
(fun z : R => cond_positivity (f z))
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) =
false
H9:cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false ->
f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0
H10:f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0 ->
cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false
H11:cond_positivity (f x) = false -> f x < 0
H12:f x < 0 -> cond_positivity (f x) = false
cond_positivity (f x) = false
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply H12.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n0 : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
Wn:=fun n0 : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n0:nat -> R
H5:(forall n0 : nat, 0 <= f (Wn n0)) -> 0 <= f x0
H6:(forall n0 : nat, f (Vn n0) <= 0) -> f x0 <= 0
H7:forall z : R, cond_positivity z = false <-> z < 0
n:nat
H8:(fun z : R => cond_positivity (f z)) x = false ->
(fun z : R => cond_positivity (f z))
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) =
false
H9:cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false ->
f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0
H10:f
  (dicho_lb x y
     (fun z : R => cond_positivity (f z)) n) < 0 ->
cond_positivity
  (f
     (dicho_lb x y
        (fun z : R => cond_positivity (f z)) n)) =
false
H11:cond_positivity (f x) = false -> f x < 0
H12:f x < 0 -> cond_positivity (f x) = false
f x < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
forall z : R, cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
cond_positivity z = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  unfold cond_positivity.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
(if Rle_dec 0 z then true else false) = false <->
z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  case (Rle_dec 0 z) as [Hle|Hnle].f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hle:0 <= z
true = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hnle:~ 0 <= z
false = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  split.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hle:0 <= z
true = false -> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hle:0 <= z
z < 0 -> true = false
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hnle:~ 0 <= z
false = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro feqt; discriminate feqt.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hle:0 <= z
z < 0 -> true = false
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hnle:~ 0 <= z
false = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ Hle H7)).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hnle:~ 0 <= z
false = false <-> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  split.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hnle:~ 0 <= z
false = false -> z < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hnle:~ 0 <= z
z < 0 -> false = false
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro; auto with real.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z0 : R => cond_positivity (f z0))) x0
p:Un_cv
  (dicho_up x y
     (fun z0 : R => cond_positivity (f z0))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
Wn:=fun n : nat =>
dicho_up x y
  (fun z0 : R => cond_positivity (f z0)) n:nat -> R
H5:(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
H6:(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
z:R
Hnle:~ 0 <= z
z < 0 -> false = false
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro; reflexivity.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut (Un_cv Wn x0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0 ->
(forall n : nat, 0 <= f (Wn n)) -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assert (H7 := continuity_seq f Wn x0 (H x0) H5).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:Un_cv (fun i : nat => f (Wn i)) (f x0)
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  destruct (total_order_T 0 (f x0)) as [[Hlt|<-]|Hgt].f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:Un_cv (fun i : nat => f (Wn i)) (f x0)
Hlt:0 < f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:Un_cv (fun i : nat => f (Wn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:Un_cv (fun i : nat => f (Wn i)) (f x0)
Hgt:0 > f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  left; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:Un_cv (fun i : nat => f (Wn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:Un_cv (fun i : nat => f (Wn i)) (f x0)
Hgt:0 > f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  right; reflexivity.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:Un_cv (fun i : nat => f (Wn i)) (f x0)
Hgt:0 > f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  unfold Un_cv in H7; unfold R_dist in H7.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut (0 < - f x0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0 -> 0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  intro.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  elim (H7 (- f x0) H8); intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut (x2 >= x2)%nat; [ intro | unfold ge; apply le_n ].f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assert (H11 := H9 x2 H10).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  rewrite Rabs_right in H11.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:f (Wn x2) - f x0 < - f x0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
f (Wn x2) - f x0 >= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  pattern (- f x0) at 1 in H11; rewrite <- Rplus_0_r in H11.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:f (Wn x2) - f x0 < - f x0 + 0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
f (Wn x2) - f x0 >= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  unfold Rminus in H11; rewrite (Rplus_comm (f (Wn x2))) in H11.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:- f x0 + f (Wn x2) < - f x0 + 0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
f (Wn x2) - f x0 >= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assert (H12 := Rplus_lt_reg_l _ _ _ H11).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:- f x0 + f (Wn x2) < - f x0 + 0
H12:f (Wn x2) < 0
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
f (Wn x2) - f x0 >= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  assert (H13 := H6 x2).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:- f x0 + f (Wn x2) < - f x0 + 0
H12:f (Wn x2) < 0
H13:0 <= f (Wn x2)
0 <= f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
f (Wn x2) - f x0 >= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H13 H12)).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
f (Wn x2) - f x0 >= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply Rle_ge; left; unfold Rminus; apply Rplus_le_lt_0_compat.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
0 <= f (Wn x2)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply H6.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
H8:0 < - f x0
x2:nat
H9:forall n : nat,
(n >= x2)%nat -> Rabs (f (Wn n) - f x0) < - f x0
H10:(x2 >= x2)%nat
H11:Rabs (f (Wn x2) - f x0) < - f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  exact H8.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Wn x0
H6:forall n : nat, 0 <= f (Wn n)
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Wn n) - f x0) < eps
Hgt:0 > f x0
0 < - f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  apply Ropp_0_gt_lt_contravar; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Wn x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  unfold Wn; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
  cut (Un_cv Vn x0).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0 ->
(forall n : nat, f (Vn n) <= 0) -> f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  assert (H7 := continuity_seq f Vn x0 (H x0) H5).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  destruct (total_order_T 0 (f x0)) as [[Hlt|<-]|Hgt].f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hlt:0 < f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  unfold Un_cv in H7; unfold R_dist in H7.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  elim (H7 (f x0) Hlt); intros.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  cut (x2 >= x2)%nat; [ intro | unfold ge; apply le_n ].f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  assert (H10 := H8 x2 H9).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  rewrite Rabs_left in H10.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:- (f (Vn x2) - f x0) < f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  pattern (f x0) at 2 in H10; rewrite <- Rplus_0_r in H10.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:- (f (Vn x2) - f x0) < f x0 + 0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  rewrite Ropp_minus_distr' in H10.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 - f (Vn x2) < f x0 + 0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  unfold Rminus in H10.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  assert (H11 := Rplus_lt_reg_l _ _ _ H10).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
H11:- f (Vn x2) < 0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  assert (H12 := H6 x2).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
H11:- f (Vn x2) < 0
H12:f (Vn x2) <= 0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  cut (0 < f (Vn x2)).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
H11:- f (Vn x2) < 0
H12:f (Vn x2) <= 0
0 < f (Vn x2) -> f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
H11:- f (Vn x2) < 0
H12:f (Vn x2) <= 0
0 < f (Vn x2)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  intro.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
H11:- f (Vn x2) < 0
H12:f (Vn x2) <= 0
H13:0 < f (Vn x2)
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
H11:- f (Vn x2) < 0
H12:f (Vn x2) <= 0
0 < f (Vn x2)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H13 H12)).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
H11:- f (Vn x2) < 0
H12:f (Vn x2) <= 0
0 < f (Vn x2)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  rewrite <- (Ropp_involutive (f (Vn x2))).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:f x0 + - f (Vn x2) < f x0 + 0
H11:- f (Vn x2) < 0
H12:f (Vn x2) <= 0
0 < - - f (Vn x2)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  apply Ropp_0_gt_lt_contravar; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f (Vn x2) - f x0 < 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  apply Rplus_lt_reg_l with (f x0 - f (Vn x2)).f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
f x0 - f (Vn x2) + (f (Vn x2) - f x0) <
f x0 - f (Vn x2) + 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  rewrite Rplus_0_r; replace (f x0 - f (Vn x2) + (f (Vn x2) - f x0)) with 0;
    [ unfold Rminus; apply Rplus_lt_le_0_compat | ring ].f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
0 < f x0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
0 <= - f (Vn x2)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (f (Vn n) - f x0) < eps
Hlt:0 < f x0
x2:nat
H8:forall n : nat,
(n >= x2)%nat -> Rabs (f (Vn n) - f x0) < f x0
H9:(x2 >= x2)%nat
H10:Rabs (f (Vn x2) - f x0) < f x0
0 <= - f (Vn x2)
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  apply Ropp_0_ge_le_contravar; apply Rle_ge; apply H6.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  right; reflexivity.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
H5:Un_cv Vn x0
H6:forall n : nat, f (Vn n) <= 0
H7:Un_cv (fun i : nat => f (Vn i)) (f x0)
Hgt:0 > f x0
f x0 <= 0
f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  left; assumption.f:R -> R
x, y:R
H:continuity f
H0:x < y
H1:f x < 0
H2:0 < f y
H3:x <= y
x1, x0:R
p0:Un_cv
  (dicho_lb x y
     (fun z : R => cond_positivity (f z))) x0
p:Un_cv
  (dicho_up x y
     (fun z : R => cond_positivity (f z))) x0
H4:x1 = x0
Vn:=fun n : nat =>
dicho_lb x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Wn:=fun n : nat =>
dicho_up x y (fun z : R => cond_positivity (f z))
  n:nat -> R
Un_cv Vn x0
  unfold Vn; assumption.
Qed.

Lemma IVT_cor :
  forall (f:R -> R) (x y:R),
    continuity f ->
    x <= y -> f x * f y <= 0 -> { z:R | x <= z <= y /\ f z = 0 }.forall (f : R -> R) (x y : R),
continuity f ->
x <= y ->
f x * f y <= 0 -> {z : R | x <= z <= y /\ f z = 0}
Proof.forall (f : R -> R) (x y : R),
continuity f ->
x <= y ->
f x * f y <= 0 -> {z : R | x <= z <= y /\ f z = 0}
  intros.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
{z : R | x <= z <= y /\ f z = 0}
  destruct (total_order_T 0 (f x)) as [[Hltx|Heqx]|Hgtx].f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  destruct (total_order_T 0 (f y)) as [[Hlty|Heqy]|Hgty].f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hlty:0 < f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  cut (0 < f x * f y);
    [ intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 H2))
      | apply Rmult_lt_0_compat; assumption ].f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  exists y.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Heqy:0 = f y
x <= y <= y /\ f y = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  split.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Heqy:0 = f y
x <= y <= y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Heqy:0 = f y
f y = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  split; [ assumption | right; reflexivity ].f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Heqy:0 = f y
f y = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  symmetry ; exact Heqy.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  cut (x < y).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y -> {z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  intro.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  assert (H3 := IVT (- f)%F x y (continuity_opp f H) H2).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  cut ((- f)%F x < 0).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0 -> {z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  cut (0 < (- f)%F y).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y ->
(- f)%F x < 0 -> {z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  intros.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
H4:0 < (- f)%F y
H5:(- f)%F x < 0
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  destruct (H3 H5 H4) as (x0,[]).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
H4:0 < (- f)%F y
H5:(- f)%F x < 0
x0:R
H6:x <= x0 <= y
H7:(- f)%F x0 = 0
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  exists x0.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
H4:0 < (- f)%F y
H5:(- f)%F x < 0
x0:R
H6:x <= x0 <= y
H7:(- f)%F x0 = 0
x <= x0 <= y /\ f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  split.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
H4:0 < (- f)%F y
H5:(- f)%F x < 0
x0:R
H6:x <= x0 <= y
H7:(- f)%F x0 = 0
x <= x0 <= y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
H4:0 < (- f)%F y
H5:(- f)%F x < 0
x0:R
H6:x <= x0 <= y
H7:(- f)%F x0 = 0
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
H4:0 < (- f)%F y
H5:(- f)%F x < 0
x0:R
H6:x <= x0 <= y
H7:(- f)%F x0 = 0
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  unfold opp_fct in H7.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
H4:0 < (- f)%F y
H5:(- f)%F x < 0
x0:R
H6:x <= x0 <= y
H7:- f x0 = 0
f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  rewrite <- (Ropp_involutive (f x0)).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
H4:0 < (- f)%F y
H5:(- f)%F x < 0
x0:R
H6:x <= x0 <= y
H7:- f x0 = 0
- - f x0 = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  apply Ropp_eq_0_compat; assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
0 < (- f)%F y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  unfold opp_fct; apply Ropp_0_gt_lt_contravar; assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
(- f)%F x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  unfold opp_fct.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
H3:(- f)%F x < 0 ->
0 < (- f)%F y ->
{z : R | x <= z <= y /\ (- f)%F z = 0}
- f x < 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  apply Rplus_lt_reg_l with (f x); rewrite Rplus_opp_r; rewrite Rplus_0_r;
    assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  inversion H0.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x < y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x = y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f x
Hgty:0 > f y
H2:x = y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  rewrite H2 in Hltx.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hltx:0 < f y
Hgty:0 > f y
H2:x = y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  elim (Rlt_irrefl _ (Rlt_trans _ _ _ Hgty Hltx)).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  exists x.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
x <= x <= y /\ f x = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  split.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
x <= x <= y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
f x = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  split; [ right; reflexivity | assumption ].f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Heqx:0 = f x
f x = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  symmetry ; assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
{z : R | x <= z <= y /\ f z = 0}
  destruct (total_order_T 0 (f y)) as [[Hlty|Heqy]|Hgty].f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  cut (x < y).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
x < y -> {z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  intro.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
H2:x < y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  apply IVT; assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  inversion H0.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
H2:x < y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
H2:x = y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hlty:0 < f y
H2:x = y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  rewrite H2 in Hgtx.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f y
Hlty:0 < f y
H2:x = y
x < y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  elim (Rlt_irrefl _ (Rlt_trans _ _ _ Hlty Hgtx)).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  exists y.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
x <= y <= y /\ f y = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  split.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
x <= y <= y
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
f y = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  split; [ assumption | right; reflexivity ].f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Heqy:0 = f y
f y = 0
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  symmetry ; assumption.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
{z : R | x <= z <= y /\ f z = 0}
  cut (0 < f x * f y).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
0 < f x * f y -> {z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
0 < f x * f y
  intro.f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
H2:0 < f x * f y
{z : R | x <= z <= y /\ f z = 0}
f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
0 < f x * f y
  elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H2 H1)).f:R -> R
x, y:R
H:continuity f
H0:x <= y
H1:f x * f y <= 0
Hgtx:0 > f x
Hgty:0 > f y
0 < f x * f y
  rewrite <- Rmult_opp_opp; apply Rmult_lt_0_compat;
    apply Ropp_0_gt_lt_contravar; assumption.
Qed.

We can now define the square root function as the reciprocal transformation of the square function

Lemma Rsqrt_exists :
  forall y:R, 0 <= y -> { z:R | 0 <= z /\ y = Rsqr z }.forall y : R, 0 <= y -> {z : R | 0 <= z /\ y = z²}
Proof.forall y : R, 0 <= y -> {z : R | 0 <= z /\ y = z²}
  intros.y:R
H:0 <= y
{z : R | 0 <= z /\ y = z²}
  set (f := fun x:R => Rsqr x - y).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
{z : R | 0 <= z /\ y = z²}
  cut (f 0 <= 0).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0 -> {z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  intro.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  cut (continuity f).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f -> {z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  intro.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  destruct (total_order_T y 1) as [[Hlt| -> ]|Hgt].y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
{z : R | 0 <= z /\ y = z²}
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  cut (0 <= f 1).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1 -> {z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  intro.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  cut (f 0 * f 1 <= 0).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0 -> {z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  intro.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  assert (X := IVT_cor f 0 1 H1 (Rlt_le _ _ Rlt_0_1) H3).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
X:{z : R | 0 <= z <= 1 /\ f z = 0}
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  elim X; intros t H4.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
X:{z : R | 0 <= z <= 1 /\ f z = 0}
t:R
H4:0 <= t <= 1 /\ f t = 0
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  exists t.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
X:{z : R | 0 <= z <= 1 /\ f z = 0}
t:R
H4:0 <= t <= 1 /\ f t = 0
0 <= t /\ y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  elim H4; intros.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
X:{z : R | 0 <= z <= 1 /\ f z = 0}
t:R
H4:0 <= t <= 1 /\ f t = 0
H5:0 <= t <= 1
H6:f t = 0
0 <= t /\ y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  split.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
X:{z : R | 0 <= z <= 1 /\ f z = 0}
t:R
H4:0 <= t <= 1 /\ f t = 0
H5:0 <= t <= 1
H6:f t = 0
0 <= t
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
X:{z : R | 0 <= z <= 1 /\ f z = 0}
t:R
H4:0 <= t <= 1 /\ f t = 0
H5:0 <= t <= 1
H6:f t = 0
y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  elim H5; intros; assumption.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
X:{z : R | 0 <= z <= 1 /\ f z = 0}
t:R
H4:0 <= t <= 1 /\ f t = 0
H5:0 <= t <= 1
H6:f t = 0
y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  unfold f in H6.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
H3:f 0 * f 1 <= 0
X:{z : R | 0 <= z <= 1 /\ f z = 0}
t:R
H4:0 <= t <= 1 /\ f t = 0
H5:0 <= t <= 1
H6:t² - y = 0
y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply Rminus_diag_uniq_sym; exact H6.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 0 * f 1 <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  rewrite Rmult_comm; pattern 0 at 2; rewrite <- (Rmult_0_r (f 1)).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
H2:0 <= f 1
f 1 * f 0 <= f 1 * 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply Rmult_le_compat_l; assumption.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= f 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  unfold f.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= 1² - y
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  rewrite Rsqr_1.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
0 <= 1 - y
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply Rplus_le_reg_l with y.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hlt:y < 1
y + 0 <= y + (1 - y)
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus;
    rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
      left; assumption.f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
{z : R | 0 <= z /\ 1 = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  exists 1.f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
0 <= 1 /\ 1 = 1²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  split.f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
0 <= 1
f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
1 = 1²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  left; apply Rlt_0_1.f:=fun x : R => x² - 1:R -> R
H1:continuity f
H0:f 0 <= 0
H:0 <= 1
1 = 1²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  symmetry; apply Rsqr_1.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  cut (0 <= f y).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y -> {z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  intro.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  cut (f 0 * f y <= 0).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0 -> {z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  intro.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  assert (X := IVT_cor f 0 y H1 H H3).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
X:{z : R | 0 <= z <= y /\ f z = 0}
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  elim X; intros t H4.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
X:{z : R | 0 <= z <= y /\ f z = 0}
t:R
H4:0 <= t <= y /\ f t = 0
{z : R | 0 <= z /\ y = z²}
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  exists t.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
X:{z : R | 0 <= z <= y /\ f z = 0}
t:R
H4:0 <= t <= y /\ f t = 0
0 <= t /\ y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  elim H4; intros.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
X:{z : R | 0 <= z <= y /\ f z = 0}
t:R
H4:0 <= t <= y /\ f t = 0
H5:0 <= t <= y
H6:f t = 0
0 <= t /\ y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  split.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
X:{z : R | 0 <= z <= y /\ f z = 0}
t:R
H4:0 <= t <= y /\ f t = 0
H5:0 <= t <= y
H6:f t = 0
0 <= t
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
X:{z : R | 0 <= z <= y /\ f z = 0}
t:R
H4:0 <= t <= y /\ f t = 0
H5:0 <= t <= y
H6:f t = 0
y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  elim H5; intros; assumption.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
X:{z : R | 0 <= z <= y /\ f z = 0}
t:R
H4:0 <= t <= y /\ f t = 0
H5:0 <= t <= y
H6:f t = 0
y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  unfold f in H6.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
H3:f 0 * f y <= 0
X:{z : R | 0 <= z <= y /\ f z = 0}
t:R
H4:0 <= t <= y /\ f t = 0
H5:0 <= t <= y
H6:t² - y = 0
y = t²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply Rminus_diag_uniq_sym; exact H6.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f 0 * f y <= 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  rewrite Rmult_comm; pattern 0 at 2; rewrite <- (Rmult_0_r (f y)).y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
H2:0 <= f y
f y * f 0 <= f y * 0
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply Rmult_le_compat_l; assumption.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= f y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  unfold f.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= y² - y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply Rplus_le_reg_l with y.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
y + 0 <= y + (y² - y)
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus;
    rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
y <= y²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  pattern y at 1; rewrite <- Rmult_1_r.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
y * 1 <= y²
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  unfold Rsqr; apply Rmult_le_compat_l.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
0 <= y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
1 <= y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  assumption.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
H1:continuity f
Hgt:y > 1
1 <= y
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  left; exact Hgt.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  replace f with (Rsqr - fct_cte y)%F.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity (Rsqr - fct_cte y)
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
(Rsqr - fct_cte y)%F = f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply continuity_minus.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity Rsqr
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity (fct_cte y)
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
(Rsqr - fct_cte y)%F = f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply derivable_continuous; apply derivable_Rsqr.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
continuity (fct_cte y)
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
(Rsqr - fct_cte y)%F = f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  apply derivable_continuous; apply derivable_const.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
H0:f 0 <= 0
(Rsqr - fct_cte y)%F = f
y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  reflexivity.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
f 0 <= 0
  unfold f; rewrite Rsqr_0.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
0 - y <= 0
  unfold Rminus; rewrite Rplus_0_l.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
- y <= 0
  apply Rge_le.y:R
H:0 <= y
f:=fun x : R => x² - y:R -> R
0 >= - y
  apply Ropp_0_le_ge_contravar; assumption.
Qed.

(* Definition of the square root: R+->R *)
Definition Rsqrt (y:nonnegreal) : R :=
  let (a,_) := Rsqrt_exists (nonneg y) (cond_nonneg y) in a.

(**********)
Lemma Rsqrt_positivity : forall x:nonnegreal, 0 <= Rsqrt x.forall x : nonnegreal, 0 <= Rsqrt x
Proof.forall x : nonnegreal, 0 <= Rsqrt x
  intro.x:nonnegreal
0 <= Rsqrt x
  destruct (Rsqrt_exists (nonneg x) (cond_nonneg x)) as (x0 & H1 & H2).x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
0 <= Rsqrt x
  cut (x0 = Rsqrt x).x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x -> 0 <= Rsqrt x
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x
  intros.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
H:x0 = Rsqrt x
0 <= Rsqrt x
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x
  rewrite <- H; assumption.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x
  unfold Rsqrt.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 =
(let (a, _) := Rsqrt_exists x (cond_nonneg x) in a)
  case (Rsqrt_exists x (cond_nonneg x)) as (?,[]).x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
x0 = x1
  apply Rsqr_inj.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
0 <= x0
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
0 <= x1
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
x0² = x1²
  assumption.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
0 <= x1
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
x0² = x1²
  assumption.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
x0² = x1²
  rewrite <- H0, <- H2; reflexivity.
Qed.

(**********)
Lemma Rsqrt_Rsqrt : forall x:nonnegreal, Rsqrt x * Rsqrt x = x.forall x : nonnegreal, Rsqrt x * Rsqrt x = x
Proof.forall x : nonnegreal, Rsqrt x * Rsqrt x = x
  intros.x:nonnegreal
Rsqrt x * Rsqrt x = x
  destruct (Rsqrt_exists (nonneg x) (cond_nonneg x)) as (x0 & H1 & H2).x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
Rsqrt x * Rsqrt x = x
  cut (x0 = Rsqrt x).x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x -> Rsqrt x * Rsqrt x = x
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x
  intros.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
H:x0 = Rsqrt x
Rsqrt x * Rsqrt x = x
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x
  rewrite <- H.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
H:x0 = Rsqrt x
x0 * x0 = x
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x
  rewrite H2; reflexivity.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 = Rsqrt x
  unfold Rsqrt.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x0 =
(let (a, _) := Rsqrt_exists x (cond_nonneg x) in a)
  case (Rsqrt_exists x (cond_nonneg x)) as (x1 & ? & ?).x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
x0 = x1
  apply Rsqr_inj.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
0 <= x0
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
0 <= x1
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
x0² = x1²
  assumption.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
0 <= x1
x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
x0² = x1²
  assumption.x:nonnegreal
x0:R
H1:0 <= x0
H2:x = x0²
x1:R
H:0 <= x1
H0:x = x1²
x0² = x1²
  rewrite <- H0, <- H2; reflexivity.
Qed.