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

Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
Require Import RList.
Require Import Classical_Prop.
Require Import Classical_Pred_Type.
Local Open Scope R_scope.

Definition included (D1 D2:R -> Prop) : Prop := forall x:R, D1 x -> D2 x.
Definition disc (x:R) (delta:posreal) (y:R) : Prop := Rabs (y - x) < delta.
Definition neighbourhood (V:R -> Prop) (x:R) : Prop :=
  exists delta : posreal, included (disc x delta) V.
Definition open_set (D:R -> Prop) : Prop :=
  forall x:R, D x -> neighbourhood D x.
Definition complementary (D:R -> Prop) (c:R) : Prop := ~ D c.
Definition closed_set (D:R -> Prop) : Prop := open_set (complementary D).
Definition intersection_domain (D1 D2:R -> Prop) (c:R) : Prop := D1 c /\ D2 c.
Definition union_domain (D1 D2:R -> Prop) (c:R) : Prop := D1 c \/ D2 c.
Definition interior (D:R -> Prop) (x:R) : Prop := neighbourhood D x.

Lemma interior_P1 : forall D:R -> Prop, included (interior D) D.
forall D : R -> Prop, included (interior D) D
Proof.
forall D : R -> Prop, included (interior D) D
  intros; unfold included; unfold interior; intros;
    unfold neighbourhood in H; elim H; intros; unfold included in H0;
      apply H0; unfold disc; unfold Rminus;
        rewrite Rplus_opp_r; rewrite Rabs_R0; apply (cond_pos x0).
Qed.

Lemma interior_P2 : forall D:R -> Prop, open_set D -> included D (interior D).
forall D : R -> Prop,
open_set D -> included D (interior D)
Proof.
forall D : R -> Prop,
open_set D -> included D (interior D)
  intros; unfold open_set in H; unfold included; intros;
    assert (H1 := H _ H0); unfold interior; apply H1.
Qed.

Definition point_adherent (D:R -> Prop) (x:R) : Prop :=
  forall V:R -> Prop,
    neighbourhood V x ->  exists y : R, intersection_domain V D y.
Definition adherence (D:R -> Prop) (x:R) : Prop := point_adherent D x.

Lemma adherence_P1 : forall D:R -> Prop, included D (adherence D).
forall D : R -> Prop, included D (adherence D)
Proof.
forall D : R -> Prop, included D (adherence D)
  intro; unfold included; intros; unfold adherence;
    unfold point_adherent; intros; exists x;
      unfold intersection_domain; split.
D:R -> Prop
x:R
H:D x
V:R -> Prop
H0:neighbourhood V x
V x
D:R -> Prop
x:R
H:D x
V:R -> Prop
H0:neighbourhood V x
D x
  unfold neighbourhood in H0; elim H0; intros; unfold included in H1; apply H1;
    unfold disc; unfold Rminus; rewrite Rplus_opp_r;
      rewrite Rabs_R0; apply (cond_pos x0).
D:R -> Prop
x:R
H:D x
V:R -> Prop
H0:neighbourhood V x
D x
  apply H.
Qed.

Lemma included_trans :
  forall D1 D2 D3:R -> Prop,
    included D1 D2 -> included D2 D3 -> included D1 D3.
forall D1 D2 D3 : R -> Prop,
included D1 D2 -> included D2 D3 -> included D1 D3
Proof.
forall D1 D2 D3 : R -> Prop,
included D1 D2 -> included D2 D3 -> included D1 D3
  unfold included; intros; apply H0; apply H; apply H1.
Qed.

Lemma interior_P3 : forall D:R -> Prop, open_set (interior D).
forall D : R -> Prop, open_set (interior D)
Proof.
forall D : R -> Prop, open_set (interior D)
  intro; unfold open_set, interior; unfold neighbourhood;
    intros; elim H; intros.
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
exists delta : posreal,
  included (disc x delta)
    (fun x1 : R =>
     exists delta0 : posreal,
       included (disc x1 delta0) D)
  exists x0; unfold included; intros.
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
exists delta : posreal,
  forall x2 : R, disc x1 delta x2 -> D x2
  set (del := x0 - Rabs (x - x1)).
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
exists delta : posreal,
  forall x2 : R, disc x1 delta x2 -> D x2
  cut (0 < del).
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del ->
exists delta : posreal,
  forall x2 : R, disc x1 delta x2 -> D x2
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  intro; exists (mkposreal del H2); intros.
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
D x2
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  cut (included (disc x1 (mkposreal del H2)) (disc x x0)).
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
included (disc x1 {| pos := del; cond_pos := H2 |})
  (disc x x0) -> D x2
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
included (disc x1 {| pos := del; cond_pos := H2 |})
  (disc x x0)
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  intro; assert (H5 := included_trans _ _ _ H4 H0).
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
H4:included
  (disc x1 {| pos := del; cond_pos := H2 |})
  (disc x x0)
H5:included
  (disc x1 {| pos := del; cond_pos := H2 |}) D
D x2
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
included (disc x1 {| pos := del; cond_pos := H2 |})
  (disc x x0)
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  apply H5; apply H3.
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
included (disc x1 {| pos := del; cond_pos := H2 |})
  (disc x x0)
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  unfold included; unfold disc; intros.
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
x3:R
H4:Rabs (x3 - x1) < {| pos := del; cond_pos := H2 |}
Rabs (x3 - x) < x0
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  apply Rle_lt_trans with (Rabs (x3 - x1) + Rabs (x1 - x)).
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
x3:R
H4:Rabs (x3 - x1) < {| pos := del; cond_pos := H2 |}
Rabs (x3 - x) <= Rabs (x3 - x1) + Rabs (x1 - x)
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
x3:R
H4:Rabs (x3 - x1) < {| pos := del; cond_pos := H2 |}
Rabs (x3 - x1) + Rabs (x1 - x) < x0
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  replace (x3 - x) with (x3 - x1 + (x1 - x)); [ apply Rabs_triang | ring ].
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
x3:R
H4:Rabs (x3 - x1) < {| pos := del; cond_pos := H2 |}
Rabs (x3 - x1) + Rabs (x1 - x) < x0
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  replace (pos x0) with (del + Rabs (x1 - x)).
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
x3:R
H4:Rabs (x3 - x1) < {| pos := del; cond_pos := H2 |}
Rabs (x3 - x1) + Rabs (x1 - x) < del + Rabs (x1 - x)
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
x3:R
H4:Rabs (x3 - x1) < {| pos := del; cond_pos := H2 |}
del + Rabs (x1 - x) = x0
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  do 2 rewrite <- (Rplus_comm (Rabs (x1 - x))); apply Rplus_lt_compat_l;
    apply H4.
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
H2:0 < del
x2:R
H3:disc x1 {| pos := del; cond_pos := H2 |} x2
x3:R
H4:Rabs (x3 - x1) < {| pos := del; cond_pos := H2 |}
del + Rabs (x1 - x) = x0
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  unfold del; rewrite <- (Rabs_Ropp (x - x1)); rewrite Ropp_minus_distr;
    ring.
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
0 < del
  unfold del; apply Rplus_lt_reg_l with (Rabs (x - x1));
    rewrite Rplus_0_r;
      replace (Rabs (x - x1) + (x0 - Rabs (x - x1))) with (pos x0);
      [ idtac | ring ].
D:R -> Prop
x:R
H:exists delta : posreal, included (disc x delta) D
x0:posreal
H0:included (disc x x0) D
x1:R
H1:disc x x0 x1
del:=x0 - Rabs (x - x1):R
Rabs (x - x1) < x0
  unfold disc in H1; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H1.
Qed.

Lemma complementary_P1 :
  forall D:R -> Prop,
    ~ (exists y : R, intersection_domain D (complementary D) y).
forall D : R -> Prop,
~
(exists y : R,
   intersection_domain D (complementary D) y)
Proof.
forall D : R -> Prop,
~
(exists y : R,
   intersection_domain D (complementary D) y)
  intro; red; intro; elim H; intros;
    unfold intersection_domain, complementary in H0; elim H0;
      intros; elim H2; assumption.
Qed.

Lemma adherence_P2 :
  forall D:R -> Prop, closed_set D -> included (adherence D) D.
forall D : R -> Prop,
closed_set D -> included (adherence D) D
Proof.
forall D : R -> Prop,
closed_set D -> included (adherence D) D
  unfold closed_set; unfold open_set, complementary; intros;
    unfold included, adherence; intros; assert (H1 := classic (D x));
      elim H1; intro.
D:R -> Prop
H:forall x0 : R,
~ D x0 -> neighbourhood (fun c : R => ~ D c) x0
x:R
H0:point_adherent D x
H1:D x \/ ~ D x
H2:D x
D x
D:R -> Prop
H:forall x0 : R,
~ D x0 -> neighbourhood (fun c : R => ~ D c) x0
x:R
H0:point_adherent D x
H1:D x \/ ~ D x
H2:~ D x
D x
  assumption.
D:R -> Prop
H:forall x0 : R,
~ D x0 -> neighbourhood (fun c : R => ~ D c) x0
x:R
H0:point_adherent D x
H1:D x \/ ~ D x
H2:~ D x
D x
  assert (H3 := H _ H2); assert (H4 := H0 _ H3); elim H4; intros;
    unfold intersection_domain in H5; elim H5; intros;
      elim H6; assumption.
Qed.

Lemma adherence_P3 : forall D:R -> Prop, closed_set (adherence D).
forall D : R -> Prop, closed_set (adherence D)
Proof.
forall D : R -> Prop, closed_set (adherence D)
  intro; unfold closed_set, adherence;
    unfold open_set, complementary, point_adherent;
      intros;
        set
          (P :=
            fun V:R -> Prop =>
              neighbourhood V x ->  exists y : R, intersection_domain V D y);
          assert (H0 := not_all_ex_not _ P H); elim H0; intros V0 H1;
            unfold P in H1; assert (H2 := imply_to_and _ _ H1);
              unfold neighbourhood; elim H2; intros; unfold neighbourhood in H3;
                elim H3; intros; exists x0; unfold included;
                  intros; red; intro.
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
False
  assert (H8 := H7 V0);
    cut (exists delta : posreal, (forall x:R, disc x1 delta x -> V0 x)).
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) -> False
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
exists delta : posreal,
  forall x2 : R, disc x1 delta x2 -> V0 x2
  intro; assert (H10 := H8 H9); elim H4; assumption.
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
exists delta : posreal,
  forall x2 : R, disc x1 delta x2 -> V0 x2
  cut (0 < x0 - Rabs (x - x1)).
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
0 < x0 - Rabs (x - x1) ->
exists delta : posreal,
  forall x2 : R, disc x1 delta x2 -> V0 x2
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
0 < x0 - Rabs (x - x1)
  intro; set (del := mkposreal _ H9); exists del; intros;
    unfold included in H5; apply H5; unfold disc;
      apply Rle_lt_trans with (Rabs (x2 - x1) + Rabs (x1 - x)).
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:forall x3 : R, disc x x0 x3 -> V0 x3
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V x3) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V0 x3) ->
exists y : R, intersection_domain V0 D y
H9:0 < x0 - Rabs (x - x1)
del:={| pos := x0 - Rabs (x - x1); cond_pos := H9 |}:posreal
x2:R
H10:disc x1 del x2
Rabs (x2 - x) <= Rabs (x2 - x1) + Rabs (x1 - x)
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:forall x3 : R, disc x x0 x3 -> V0 x3
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V x3) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V0 x3) ->
exists y : R, intersection_domain V0 D y
H9:0 < x0 - Rabs (x - x1)
del:={| pos := x0 - Rabs (x - x1); cond_pos := H9 |}:posreal
x2:R
H10:disc x1 del x2
Rabs (x2 - x1) + Rabs (x1 - x) < x0
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
0 < x0 - Rabs (x - x1)
  replace (x2 - x) with (x2 - x1 + (x1 - x)); [ apply Rabs_triang | ring ].
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:forall x3 : R, disc x x0 x3 -> V0 x3
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V x3) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V0 x3) ->
exists y : R, intersection_domain V0 D y
H9:0 < x0 - Rabs (x - x1)
del:={| pos := x0 - Rabs (x - x1); cond_pos := H9 |}:posreal
x2:R
H10:disc x1 del x2
Rabs (x2 - x1) + Rabs (x1 - x) < x0
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
0 < x0 - Rabs (x - x1)
  replace (pos x0) with (del + Rabs (x1 - x)).
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:forall x3 : R, disc x x0 x3 -> V0 x3
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V x3) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V0 x3) ->
exists y : R, intersection_domain V0 D y
H9:0 < x0 - Rabs (x - x1)
del:={| pos := x0 - Rabs (x - x1); cond_pos := H9 |}:posreal
x2:R
H10:disc x1 del x2
Rabs (x2 - x1) + Rabs (x1 - x) < del + Rabs (x1 - x)
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:forall x3 : R, disc x x0 x3 -> V0 x3
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V x3) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V0 x3) ->
exists y : R, intersection_domain V0 D y
H9:0 < x0 - Rabs (x - x1)
del:={| pos := x0 - Rabs (x - x1); cond_pos := H9 |}:posreal
x2:R
H10:disc x1 del x2
del + Rabs (x1 - x) = x0
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
0 < x0 - Rabs (x - x1)
  do 2 rewrite <- (Rplus_comm (Rabs (x1 - x))); apply Rplus_lt_compat_l;
    apply H10.
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:forall x3 : R, disc x x0 x3 -> V0 x3
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V x3) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x3 : R, disc x1 delta x3 -> V0 x3) ->
exists y : R, intersection_domain V0 D y
H9:0 < x0 - Rabs (x - x1)
del:={| pos := x0 - Rabs (x - x1); cond_pos := H9 |}:posreal
x2:R
H10:disc x1 del x2
del + Rabs (x1 - x) = x0
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
0 < x0 - Rabs (x - x1)
  unfold del; simpl; rewrite <- (Rabs_Ropp (x - x1));
    rewrite Ropp_minus_distr; ring.
D:R -> Prop
x:R
H:~
(forall V : R -> Prop,
 neighbourhood V x ->
 exists y : R, intersection_domain V D y)
P:=fun V : R -> Prop =>
neighbourhood V x ->
exists y : R, intersection_domain V D y:(R -> Prop) -> Prop
H0:exists n : R -> Prop, ~ P n
V0:R -> Prop
H1:~
(neighbourhood V0 x ->
 exists y : R, intersection_domain V0 D y)
H2:neighbourhood V0 x /\
~ (exists y : R, intersection_domain V0 D y)
H3:exists delta : posreal,
  included (disc x delta) V0
H4:~ (exists y : R, intersection_domain V0 D y)
x0:posreal
H5:included (disc x x0) V0
x1:R
H6:disc x x0 x1
H7:forall V : R -> Prop,
(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V x2) ->
exists y : R, intersection_domain V D y
H8:(exists delta : posreal,
   forall x2 : R, disc x1 delta x2 -> V0 x2) ->
exists y : R, intersection_domain V0 D y
0 < x0 - Rabs (x - x1)
  apply Rplus_lt_reg_l with (Rabs (x - x1)); rewrite Rplus_0_r;
    replace (Rabs (x - x1) + (x0 - Rabs (x - x1))) with (pos x0);
    [ rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H6 | ring ].
Qed.

Definition eq_Dom (D1 D2:R -> Prop) : Prop :=
  included D1 D2 /\ included D2 D1.

Infix "=_D" := eq_Dom (at level 70, no associativity).

Lemma open_set_P1 : forall D:R -> Prop, open_set D <-> D =_D interior D.
forall D : R -> Prop, open_set D <-> D =_D interior D
Proof.
forall D : R -> Prop, open_set D <-> D =_D interior D
  intro; split.
D:R -> Prop
open_set D -> D =_D interior D
D:R -> Prop
D =_D interior D -> open_set D
  intro; unfold eq_Dom; split.
D:R -> Prop
H:open_set D
included D (interior D)
D:R -> Prop
H:open_set D
included (interior D) D
D:R -> Prop
D =_D interior D -> open_set D
  apply interior_P2; assumption.
D:R -> Prop
H:open_set D
included (interior D) D
D:R -> Prop
D =_D interior D -> open_set D
  apply interior_P1.
D:R -> Prop
D =_D interior D -> open_set D
  intro; unfold eq_Dom in H; elim H; clear H; intros; unfold open_set;
    intros; unfold included, interior in H; unfold included in H0;
      apply (H _ H1).
Qed.

Lemma closed_set_P1 : forall D:R -> Prop, closed_set D <-> D =_D adherence D.
forall D : R -> Prop,
closed_set D <-> D =_D adherence D
Proof.
forall D : R -> Prop,
closed_set D <-> D =_D adherence D
  intro; split.
D:R -> Prop
closed_set D -> D =_D adherence D
D:R -> Prop
D =_D adherence D -> closed_set D
  intro; unfold eq_Dom; split.
D:R -> Prop
H:closed_set D
included D (adherence D)
D:R -> Prop
H:closed_set D
included (adherence D) D
D:R -> Prop
D =_D adherence D -> closed_set D
  apply adherence_P1.
D:R -> Prop
H:closed_set D
included (adherence D) D
D:R -> Prop
D =_D adherence D -> closed_set D
  apply adherence_P2; assumption.
D:R -> Prop
D =_D adherence D -> closed_set D
  unfold eq_Dom; unfold included; intros;
    assert (H0 := adherence_P3 D); unfold closed_set in H0;
      unfold closed_set; unfold open_set;
        unfold open_set in H0; intros; assert (H2 : complementary (adherence D) x).
D:R -> Prop
H:(forall x0 : R, D x0 -> adherence D x0) /\
(forall x0 : R, adherence D x0 -> D x0)
H0:forall x0 : R,
complementary (adherence D) x0 ->
neighbourhood (complementary (adherence D)) x0
x:R
H1:complementary D x
complementary (adherence D) x
D:R -> Prop
H:(forall x0 : R, D x0 -> adherence D x0) /\
(forall x0 : R, adherence D x0 -> D x0)
H0:forall x0 : R,
complementary (adherence D) x0 ->
neighbourhood (complementary (adherence D)) x0
x:R
H1:complementary D x
H2:complementary (adherence D) x
neighbourhood (complementary D) x
  unfold complementary; unfold complementary in H1; red; intro;
    elim H; clear H; intros _ H; elim H1; apply (H _ H2).
D:R -> Prop
H:(forall x0 : R, D x0 -> adherence D x0) /\
(forall x0 : R, adherence D x0 -> D x0)
H0:forall x0 : R,
complementary (adherence D) x0 ->
neighbourhood (complementary (adherence D)) x0
x:R
H1:complementary D x
H2:complementary (adherence D) x
neighbourhood (complementary D) x
  assert (H3 := H0 _ H2); unfold neighbourhood;
    unfold neighbourhood in H3; elim H3; intros; exists x0;
      unfold included; unfold included in H4; intros;
        assert (H6 := H4 _ H5); unfold complementary in H6;
          unfold complementary; red; intro;
            elim H; clear H; intros H _; elim H6; apply (H _ H7).
Qed.

Lemma neighbourhood_P1 :
  forall (D1 D2:R -> Prop) (x:R),
    included D1 D2 -> neighbourhood D1 x -> neighbourhood D2 x.
forall (D1 D2 : R -> Prop) (x : R),
included D1 D2 ->
neighbourhood D1 x -> neighbourhood D2 x
Proof.
forall (D1 D2 : R -> Prop) (x : R),
included D1 D2 ->
neighbourhood D1 x -> neighbourhood D2 x
  unfold included, neighbourhood; intros; elim H0; intros; exists x0;
    intros; unfold included; unfold included in H1;
      intros; apply (H _ (H1 _ H2)).
Qed.

Lemma open_set_P2 :
  forall D1 D2:R -> Prop,
    open_set D1 -> open_set D2 -> open_set (union_domain D1 D2).
forall D1 D2 : R -> Prop,
open_set D1 ->
open_set D2 -> open_set (union_domain D1 D2)
Proof.
forall D1 D2 : R -> Prop,
open_set D1 ->
open_set D2 -> open_set (union_domain D1 D2)
  unfold open_set; intros; unfold union_domain in H1; elim H1; intro.
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D1 x
neighbourhood (union_domain D1 D2) x
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D2 x
neighbourhood (union_domain D1 D2) x
  apply neighbourhood_P1 with D1.
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D1 x
included D1 (union_domain D1 D2)
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D1 x
neighbourhood D1 x
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D2 x
neighbourhood (union_domain D1 D2) x
  unfold included, union_domain; tauto.
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D1 x
neighbourhood D1 x
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D2 x
neighbourhood (union_domain D1 D2) x
  apply H; assumption.
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D2 x
neighbourhood (union_domain D1 D2) x
  apply neighbourhood_P1 with D2.
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D2 x
included D2 (union_domain D1 D2)
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D2 x
neighbourhood D2 x
  unfold included, union_domain; tauto.
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x \/ D2 x
H2:D2 x
neighbourhood D2 x
  apply H0; assumption.
Qed.

Lemma open_set_P3 :
  forall D1 D2:R -> Prop,
    open_set D1 -> open_set D2 -> open_set (intersection_domain D1 D2).
forall D1 D2 : R -> Prop,
open_set D1 ->
open_set D2 -> open_set (intersection_domain D1 D2)
Proof.
forall D1 D2 : R -> Prop,
open_set D1 ->
open_set D2 -> open_set (intersection_domain D1 D2)
  unfold open_set; intros; unfold intersection_domain in H1; elim H1;
    intros.
D1, D2:R -> Prop
H:forall x0 : R, D1 x0 -> neighbourhood D1 x0
H0:forall x0 : R, D2 x0 -> neighbourhood D2 x0
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
neighbourhood (intersection_domain D1 D2) x
  assert (H4 := H _ H2); assert (H5 := H0 _ H3);
    unfold intersection_domain; unfold neighbourhood in H4, H5;
      elim H4; clear H; intros del1 H; elim H5; clear H0;
        intros del2 H0; cut (0 < Rmin del1 del2).
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2 ->
neighbourhood (fun c : R => D1 c /\ D2 c) x
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  intro; set (del := mkposreal _ H6).
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
neighbourhood (fun c : R => D1 c /\ D2 c) x
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  exists del; unfold included; intros; unfold included in H, H0;
    unfold disc in H, H0, H7.
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
D1 x0 /\ D2 x0
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  split.
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
D1 x0
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
D2 x0
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  apply H; apply Rlt_le_trans with (pos del).
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
Rabs (x0 - x) < del
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
del <= del1
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
D2 x0
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  apply H7.
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
del <= del1
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
D2 x0
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  unfold del; simpl; apply Rmin_l.
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
D2 x0
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  apply H0; apply Rlt_le_trans with (pos del).
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
Rabs (x0 - x) < del
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
del <= del2
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  apply H7.
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:forall x1 : R, Rabs (x1 - x) < del1 -> D1 x1
del2:posreal
H0:forall x1 : R, Rabs (x1 - x) < del2 -> D2 x1
H6:0 < Rmin del1 del2
del:={| pos := Rmin del1 del2; cond_pos := H6 |}:posreal
x0:R
H7:Rabs (x0 - x) < del
del <= del2
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  unfold del; simpl; apply Rmin_r.
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
0 < Rmin del1 del2
  unfold Rmin; case (Rle_dec del1 del2); intro.
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
r:del1 <= del2
0 < del1
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
n:~ del1 <= del2
0 < del2
  apply (cond_pos del1).
D1, D2:R -> Prop
x:R
H1:D1 x /\ D2 x
H2:D1 x
H3:D2 x
H4:exists delta : posreal,
  included (disc x delta) D1
H5:exists delta : posreal,
  included (disc x delta) D2
del1:posreal
H:included (disc x del1) D1
del2:posreal
H0:included (disc x del2) D2
n:~ del1 <= del2
0 < del2
  apply (cond_pos del2).
Qed.

Lemma open_set_P4 : open_set (fun x:R => False).
open_set (fun _ : R => False)
Proof.
open_set (fun _ : R => False)
  unfold open_set; intros; elim H.
Qed.

Lemma open_set_P5 : open_set (fun x:R => True).
open_set (fun _ : R => True)
Proof.
open_set (fun _ : R => True)
  unfold open_set; intros; unfold neighbourhood.
x:R
H:True
exists delta : posreal,
  included (disc x delta) (fun _ : R => True)
  exists (mkposreal 1 Rlt_0_1); unfold included; intros; trivial.
Qed.

Lemma disc_P1 : forall (x:R) (del:posreal), open_set (disc x del).
forall (x : R) (del : posreal), open_set (disc x del)
Proof.
forall (x : R) (del : posreal), open_set (disc x del)
  intros; assert (H := open_set_P1 (disc x del)).
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
open_set (disc x del)
  elim H; intros; apply H1.
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
disc x del =_D interior (disc x del)
  unfold eq_Dom; split.
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (disc x del) (interior (disc x del))
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  unfold included, interior, disc; intros;
    cut (0 < del - Rabs (x - x0)).
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0) ->
neighbourhood (fun y : R => Rabs (y - x) < del) x0
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  intro; set (del2 := mkposreal _ H3).
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
neighbourhood (fun y : R => Rabs (y - x) < del) x0
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  exists del2; unfold included; intros.
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
Rabs (x1 - x) < del
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  apply Rle_lt_trans with (Rabs (x1 - x0) + Rabs (x0 - x)).
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
Rabs (x1 - x) <= Rabs (x1 - x0) + Rabs (x0 - x)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
Rabs (x1 - x0) + Rabs (x0 - x) < del
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  replace (x1 - x) with (x1 - x0 + (x0 - x)); [ apply Rabs_triang | ring ].
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
Rabs (x1 - x0) + Rabs (x0 - x) < del
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  replace (pos del) with (del2 + Rabs (x0 - x)).
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
Rabs (x1 - x0) + Rabs (x0 - x) < del2 + Rabs (x0 - x)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
del2 + Rabs (x0 - x) = del
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  do 2 rewrite <- (Rplus_comm (Rabs (x0 - x))); apply Rplus_lt_compat_l.
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
Rabs (x1 - x0) < del2
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
del2 + Rabs (x0 - x) = del
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  apply H4.
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
H3:0 < del - Rabs (x - x0)
del2:={| pos := del - Rabs (x - x0); cond_pos := H3 |}:posreal
x1:R
H4:disc x0 del2 x1
del2 + Rabs (x0 - x) = del
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  unfold del2; simpl; rewrite <- (Rabs_Ropp (x - x0));
    rewrite Ropp_minus_distr; ring.
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
x0:R
H2:Rabs (x0 - x) < del
0 < del - Rabs (x - x0)
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  apply Rplus_lt_reg_l with (Rabs (x - x0)); rewrite Rplus_0_r;
    replace (Rabs (x - x0) + (del - Rabs (x - x0))) with (pos del);
    [ rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H2 | ring ].
x:R
del:posreal
H:open_set (disc x del) <->
disc x del =_D interior (disc x del)
H0:open_set (disc x del) ->
disc x del =_D interior (disc x del)
H1:disc x del =_D interior (disc x del) ->
open_set (disc x del)
included (interior (disc x del)) (disc x del)
  apply interior_P1.
Qed.

Lemma continuity_P1 :
  forall (f:R -> R) (x:R),
    continuity_pt f x <->
    (forall W:R -> Prop,
      neighbourhood W (f x) ->
      exists V : R -> Prop,
        neighbourhood V x /\ (forall y:R, V y -> W (f y))).
forall (f : R -> R) (x : R),
continuity_pt f x <->
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y)))
Proof.
forall (f : R -> R) (x : R),
continuity_pt f x <->
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y)))
  intros; split.
f:R -> R
x:R
continuity_pt f x ->
forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\ (forall y : R, V y -> W (f y))
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  intros; unfold neighbourhood in H0.
f:R -> R
x:R
H:continuity_pt f x
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
exists V : R -> Prop,
  neighbourhood V x /\ (forall y : R, V y -> W (f y))
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  elim H0; intros del1 H1.
f:R -> R
x:R
H:continuity_pt f x
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:included (disc (f x) del1) W
exists V : R -> Prop,
  neighbourhood V x /\ (forall y : R, V y -> W (f y))
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  unfold continuity_pt in H; unfold continue_in in H; unfold limit1_in in H;
    unfold limit_in in H; simpl in H; unfold R_dist in H.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:included (disc (f x) del1) W
exists V : R -> Prop,
  neighbourhood V x /\ (forall y : R, V y -> W (f y))
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  assert (H2 := H del1 (cond_pos del1)).
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:included (disc (f x) del1) W
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
exists V : R -> Prop,
  neighbourhood V x /\ (forall y : R, V y -> W (f y))
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  elim H2; intros del2 H3.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:included (disc (f x) del1) W
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
exists V : R -> Prop,
  neighbourhood V x /\ (forall y : R, V y -> W (f y))
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  elim H3; intros.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:included (disc (f x) del1) W
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
exists V : R -> Prop,
  neighbourhood V x /\ (forall y : R, V y -> W (f y))
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  exists (disc x (mkposreal del2 H4)).
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:included (disc (f x) del1) W
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
neighbourhood
  (disc x {| pos := del2; cond_pos := H4 |}) x /\
(forall y : R,
 disc x {| pos := del2; cond_pos := H4 |} y -> W (f y))
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  intros; unfold included in H1; split.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
neighbourhood
  (disc x {| pos := del2; cond_pos := H4 |}) x
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
forall y : R,
disc x {| pos := del2; cond_pos := H4 |} y -> W (f y)
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  unfold neighbourhood, disc.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
exists delta : posreal,
  included (fun y : R => Rabs (y - x) < delta)
    (fun y : R =>
     Rabs (y - x) < {| pos := del2; cond_pos := H4 |})
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
forall y : R,
disc x {| pos := del2; cond_pos := H4 |} y -> W (f y)
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  exists (mkposreal del2 H4).
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
included
  (fun y : R =>
   Rabs (y - x) < {| pos := del2; cond_pos := H4 |})
  (fun y : R =>
   Rabs (y - x) < {| pos := del2; cond_pos := H4 |})
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
forall y : R,
disc x {| pos := del2; cond_pos := H4 |} y -> W (f y)
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  unfold included; intros; assumption.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
forall y : R,
disc x {| pos := del2; cond_pos := H4 |} y -> W (f y)
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  intros; apply H1; unfold disc; case (Req_dec y x); intro.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y = x
Rabs (f y - f x) < del1
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
Rabs (f y - f x) < del1
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  rewrite H7; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
    apply (cond_pos del1).
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
Rabs (f y - f x) < del1
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  apply H5; split.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
D_x no_cond x y
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
Rabs (y - x) < del2
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  unfold D_x, no_cond; split.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
True
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
x <> y
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
Rabs (y - x) < del2
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  trivial.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
x <> y
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
Rabs (y - x) < del2
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  apply (not_eq_sym (A:=R)); apply H7.
f:R -> R
x:R
H:forall eps : R,
eps > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
W:R -> Prop
H0:exists delta : posreal,
  included (disc (f x) delta) W
del1:posreal
H1:forall x0 : R, disc (f x) del1 x0 -> W x0
H2:exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < del1)
del2:R
H3:del2 > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
 Rabs (f x0 - f x) < del1)
H4:del2 > 0
H5:forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del2 ->
Rabs (f x0 - f x) < del1
y:R
H6:disc x {| pos := del2; cond_pos := H4 |} y
H7:y <> x
Rabs (y - x) < del2
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  unfold disc in H6; apply H6.
f:R -> R
x:R
(forall W : R -> Prop,
 neighbourhood W (f x) ->
 exists V : R -> Prop,
   neighbourhood V x /\ (forall y : R, V y -> W (f y))) ->
continuity_pt f x
  intros; unfold continuity_pt; unfold continue_in;
    unfold limit1_in; unfold limit_in;
      intros.
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) (f x) < eps)
  assert (H1 := H (disc (f x) (mkposreal eps H0))).
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) (f x) < eps)
  cut (neighbourhood (disc (f x) (mkposreal eps H0)) (f x)).
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |}) 
  (f x) ->
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) (f x) < eps)
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |}) 
  (f x)
  intro; assert (H3 := H1 H2).
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
H2:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x)
H3:exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) (f x) < eps)
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |}) 
  (f x)
  elim H3; intros D H4; elim H4; intros; unfold neighbourhood in H5; elim H5;
    intros del1 H7.
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
H2:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x)
H3:exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
D:R -> Prop
H4:neighbourhood D x /\
(forall y : R,
 D y ->
 disc (f x) {| pos := eps; cond_pos := H0 |}
   (f y))
H5:exists delta : posreal, included (disc x delta) D
H6:forall y : R,
D y ->
disc (f x) {| pos := eps; cond_pos := H0 |} (f y)
del1:posreal
H7:included (disc x del1) D
exists alp : R,
  alp > 0 /\
  (forall x0 : Base R_met,
   D_x no_cond x x0 /\ dist R_met x0 x < alp ->
   dist R_met (f x0) (f x) < eps)
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |}) 
  (f x)
  exists (pos del1); split.
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
H2:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x)
H3:exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
D:R -> Prop
H4:neighbourhood D x /\
(forall y : R,
 D y ->
 disc (f x) {| pos := eps; cond_pos := H0 |}
   (f y))
H5:exists delta : posreal, included (disc x delta) D
H6:forall y : R,
D y ->
disc (f x) {| pos := eps; cond_pos := H0 |} (f y)
del1:posreal
H7:included (disc x del1) D
del1 > 0
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
H2:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x)
H3:exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
D:R -> Prop
H4:neighbourhood D x /\
(forall y : R,
 D y ->
 disc (f x) {| pos := eps; cond_pos := H0 |}
   (f y))
H5:exists delta : posreal, included (disc x delta) D
H6:forall y : R,
D y ->
disc (f x) {| pos := eps; cond_pos := H0 |} (f y)
del1:posreal
H7:included (disc x del1) D
forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < del1 ->
dist R_met (f x0) (f x) < eps
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |}) 
  (f x)
  apply (cond_pos del1).
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
H2:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x)
H3:exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
D:R -> Prop
H4:neighbourhood D x /\
(forall y : R,
 D y ->
 disc (f x) {| pos := eps; cond_pos := H0 |}
   (f y))
H5:exists delta : posreal, included (disc x delta) D
H6:forall y : R,
D y ->
disc (f x) {| pos := eps; cond_pos := H0 |} (f y)
del1:posreal
H7:included (disc x del1) D
forall x0 : Base R_met,
D_x no_cond x x0 /\ dist R_met x0 x < del1 ->
dist R_met (f x0) (f x) < eps
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |}) 
  (f x)
  intros; elim H8; intros; simpl in H10; unfold R_dist in H10; simpl;
    unfold R_dist; apply (H6 _ (H7 _ H10)).
f:R -> R
x:R
H:forall W : R -> Prop,
neighbourhood W (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R, V y -> W (f y))
eps:R
H0:eps > 0
H1:neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |})
  (f x) ->
exists V : R -> Prop,
  neighbourhood V x /\
  (forall y : R,
   V y ->
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f y))
neighbourhood
  (disc (f x) {| pos := eps; cond_pos := H0 |}) 
  (f x)
  unfold neighbourhood, disc; exists (mkposreal eps H0);
    unfold included; intros; assumption.
Qed.

Definition image_rec (f:R -> R) (D:R -> Prop) (x:R) : Prop := D (f x).

(**********)
Lemma continuity_P2 :
  forall (f:R -> R) (D:R -> Prop),
    continuity f -> open_set D -> open_set (image_rec f D).
forall (f : R -> R) (D : R -> Prop),
continuity f -> open_set D -> open_set (image_rec f D)
Proof.
forall (f : R -> R) (D : R -> Prop),
continuity f -> open_set D -> open_set (image_rec f D)
  intros; unfold open_set in H0; unfold open_set; intros;
    assert (H2 := continuity_P1 f x); elim H2; intros H3 _;
      assert (H4 := H3 (H x)); unfold neighbourhood, image_rec;
        unfold image_rec in H1; assert (H5 := H4 D (H0 (f x) H1));
          elim H5; intros V0 H6; elim H6; intros; unfold neighbourhood in H7;
            elim H7; intros del H9; exists del; unfold included in H9;
              unfold included; intros; apply (H8 _ (H9 _ H10)).
Qed.

(**********)
Lemma continuity_P3 :
  forall f:R -> R,
    continuity f <->
    (forall D:R -> Prop, open_set D -> open_set (image_rec f D)).
forall f : R -> R,
continuity f <->
(forall D : R -> Prop,
 open_set D -> open_set (image_rec f D))
Proof.
forall f : R -> R,
continuity f <->
(forall D : R -> Prop,
 open_set D -> open_set (image_rec f D))
  intros; split.
f:R -> R
continuity f ->
forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
f:R -> R
(forall D : R -> Prop,
 open_set D -> open_set (image_rec f D)) ->
continuity f
  intros; apply continuity_P2; assumption.
f:R -> R
(forall D : R -> Prop,
 open_set D -> open_set (image_rec f D)) ->
continuity f
  intros; unfold continuity; unfold continuity_pt;
    unfold continue_in; unfold limit1_in;
      unfold limit_in; simpl; unfold R_dist;
        intros; cut (open_set (disc (f x) (mkposreal _ H0))).
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |}) ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  intro; assert (H2 := H _ H1).
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:open_set
  (image_rec f
     (disc (f x) {| pos := eps; cond_pos := H0 |}))
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  unfold open_set, image_rec in H2; cut (disc (f x) (mkposreal _ H0) (f x)).
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
disc (f x) {| pos := eps; cond_pos := H0 |} (f x) ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  intro; assert (H4 := H2 _ H3).
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
H3:disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
H4:neighbourhood
  (fun x0 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x0)) x
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  unfold neighbourhood in H4; elim H4; intros del H5.
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
H3:disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
H4:exists delta : posreal,
  included (disc x delta)
    (fun x0 : R =>
     disc (f x) {| pos := eps; cond_pos := H0 |}
       (f x0))
del:posreal
H5:included (disc x del)
  (fun x0 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x0))
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (f x0 - f x) < eps)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  exists (pos del); split.
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
H3:disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
H4:exists delta : posreal,
  included (disc x delta)
    (fun x0 : R =>
     disc (f x) {| pos := eps; cond_pos := H0 |}
       (f x0))
del:posreal
H5:included (disc x del)
  (fun x0 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x0))
del > 0
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
H3:disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
H4:exists delta : posreal,
  included (disc x delta)
    (fun x0 : R =>
     disc (f x) {| pos := eps; cond_pos := H0 |}
       (f x0))
del:posreal
H5:included (disc x del)
  (fun x0 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x0))
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del ->
Rabs (f x0 - f x) < eps
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  apply (cond_pos del).
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
H3:disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
H4:exists delta : posreal,
  included (disc x delta)
    (fun x0 : R =>
     disc (f x) {| pos := eps; cond_pos := H0 |}
       (f x0))
del:posreal
H5:included (disc x del)
  (fun x0 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x0))
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < del ->
Rabs (f x0 - f x) < eps
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  intros; unfold included in H5; apply H5; elim H6; intros; apply H8.
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
H1:open_set
  (disc (f x) {| pos := eps; cond_pos := H0 |})
H2:forall x0 : R,
disc (f x) {| pos := eps; cond_pos := H0 |}
  (f x0) ->
neighbourhood
  (fun x1 : R =>
   disc (f x) {| pos := eps; cond_pos := H0 |}
     (f x1)) x0
disc (f x) {| pos := eps; cond_pos := H0 |} (f x)
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  unfold disc; unfold Rminus; rewrite Rplus_opp_r;
    rewrite Rabs_R0; apply H0.
f:R -> R
H:forall D : R -> Prop,
open_set D -> open_set (image_rec f D)
x, eps:R
H0:eps > 0
open_set (disc (f x) {| pos := eps; cond_pos := H0 |})
  apply disc_P1.
Qed.

(**********)
Theorem Rsepare :
  forall x y:R,
    x <> y ->
    exists V : R -> Prop,
      (exists W : R -> Prop,
        neighbourhood V x /\
        neighbourhood W y /\ ~ (exists y : R, intersection_domain V W y)).
forall x y : R,
x <> y ->
exists V W : R -> Prop,
  neighbourhood V x /\
  neighbourhood W y /\
  ~ (exists y0 : R, intersection_domain V W y0)
Proof.
forall x y : R,
x <> y ->
exists V W : R -> Prop,
  neighbourhood V x /\
  neighbourhood W y /\
  ~ (exists y0 : R, intersection_domain V W y0)
  intros x y Hsep; set (D := Rabs (x - y)).
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
exists V W : R -> Prop,
  neighbourhood V x /\
  neighbourhood W y /\
  ~ (exists y0 : R, intersection_domain V W y0)
  cut (0 < D / 2).
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2 ->
exists V W : R -> Prop,
  neighbourhood V x /\
  neighbourhood W y /\
  ~ (exists y0 : R, intersection_domain V W y0)
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  intro; exists (disc x (mkposreal _ H)).
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
exists W : R -> Prop,
  neighbourhood
    (disc x {| pos := D / 2; cond_pos := H |}) x /\
  neighbourhood W y /\
  ~
  (exists y0 : R,
     intersection_domain
       (disc x {| pos := D / 2; cond_pos := H |}) W y0)
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  exists (disc y (mkposreal _ H)); split.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
neighbourhood
  (disc x {| pos := D / 2; cond_pos := H |}) x
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
neighbourhood
  (disc y {| pos := D / 2; cond_pos := H |}) y /\
~
(exists y0 : R,
   intersection_domain
     (disc x {| pos := D / 2; cond_pos := H |})
     (disc y {| pos := D / 2; cond_pos := H |}) y0)
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  unfold neighbourhood; exists (mkposreal _ H); unfold included;
    tauto.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
neighbourhood
  (disc y {| pos := D / 2; cond_pos := H |}) y /\
~
(exists y0 : R,
   intersection_domain
     (disc x {| pos := D / 2; cond_pos := H |})
     (disc y {| pos := D / 2; cond_pos := H |}) y0)
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  split.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
neighbourhood
  (disc y {| pos := D / 2; cond_pos := H |}) y
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
~
(exists y0 : R,
   intersection_domain
     (disc x {| pos := D / 2; cond_pos := H |})
     (disc y {| pos := D / 2; cond_pos := H |}) y0)
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  unfold neighbourhood; exists (mkposreal _ H); unfold included;
    tauto.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
~
(exists y0 : R,
   intersection_domain
     (disc x {| pos := D / 2; cond_pos := H |})
     (disc y {| pos := D / 2; cond_pos := H |}) y0)
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  red; intro; elim H0; intros; unfold intersection_domain in H1;
    elim H1; intros.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
False
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  cut (D < D).
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
D < D -> False
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
D < D
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  intro; elim (Rlt_irrefl _ H4).
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
D < D
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  change (Rabs (x - y) < D);
    apply Rle_lt_trans with (Rabs (x - x0) + Rabs (x0 - y)).
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
Rabs (x - y) <= Rabs (x - x0) + Rabs (x0 - y)
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
Rabs (x - x0) + Rabs (x0 - y) < D
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  replace (x - y) with (x - x0 + (x0 - y)); [ apply Rabs_triang | ring ].
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
Rabs (x - x0) + Rabs (x0 - y) < D
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  rewrite (double_var D); apply Rplus_lt_compat.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
Rabs (x - x0) < D / 2
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
Rabs (x0 - y) < D / 2
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H2.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
H:0 < D / 2
H0:exists y0 : R,
  intersection_domain
    (disc x {| pos := D / 2; cond_pos := H |})
    (disc y {| pos := D / 2; cond_pos := H |}) y0
x0:R
H1:disc x {| pos := D / 2; cond_pos := H |} x0 /\
disc y {| pos := D / 2; cond_pos := H |} x0
H2:disc x {| pos := D / 2; cond_pos := H |} x0
H3:disc y {| pos := D / 2; cond_pos := H |} x0
Rabs (x0 - y) < D / 2
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  apply H3.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D / 2
  unfold Rdiv; apply Rmult_lt_0_compat.
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < D
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < / 2
  unfold D; apply Rabs_pos_lt; apply (Rminus_eq_contra _ _ Hsep).
x, y:R
Hsep:x <> y
D:=Rabs (x - y):R
0 < / 2
  apply Rinv_0_lt_compat; prove_sup0.
Qed.

Record family : Type := mkfamily
  {ind : R -> Prop;
    f :> R -> R -> Prop;
    cond_fam : forall x:R, (exists y : R, f x y) -> ind x}.

Definition family_open_set (f:family) : Prop := forall x:R, open_set (f x).

Definition domain_finite (D:R -> Prop) : Prop :=
  exists l : Rlist, (forall x:R, D x <-> In x l).

Definition family_finite (f:family) : Prop := domain_finite (ind f).

Definition covering (D:R -> Prop) (f:family) : Prop :=
  forall x:R, D x ->  exists y : R, f y x.

Definition covering_open_set (D:R -> Prop) (f:family) : Prop :=
  covering D f /\ family_open_set f.

Definition covering_finite (D:R -> Prop) (f:family) : Prop :=
  covering D f /\ family_finite f.

Lemma restriction_family :
  forall (f:family) (D:R -> Prop) (x:R),
    (exists y : R, (fun z1 z2:R => f z1 z2 /\ D z1) x y) ->
    intersection_domain (ind f) D x.
forall (f0 : family) (D : R -> Prop) (x : R),
(exists y : R, (fun z1 z2 : R => f0 z1 z2 /\ D z1) x y) ->
intersection_domain (ind f0) D x
Proof.
forall (f0 : family) (D : R -> Prop) (x : R),
(exists y : R, (fun z1 z2 : R => f0 z1 z2 /\ D z1) x y) ->
intersection_domain (ind f0) D x
  intros; elim H; intros; unfold intersection_domain; elim H0; intros;
    split.
f0:family
D:R -> Prop
x:R
H:exists y : R,
  (fun z1 z2 : R => f0 z1 z2 /\ D z1) x y
x0:R
H0:f0 x x0 /\ D x
H1:f0 x x0
H2:D x
ind f0 x
f0:family
D:R -> Prop
x:R
H:exists y : R,
  (fun z1 z2 : R => f0 z1 z2 /\ D z1) x y
x0:R
H0:f0 x x0 /\ D x
H1:f0 x x0
H2:D x
D x
  apply (cond_fam f0); exists x0; assumption.
f0:family
D:R -> Prop
x:R
H:exists y : R,
  (fun z1 z2 : R => f0 z1 z2 /\ D z1) x y
x0:R
H0:f0 x x0 /\ D x
H1:f0 x x0
H2:D x
D x
  assumption.
Qed.

Definition subfamily (f:family) (D:R -> Prop) : family :=
  mkfamily (intersection_domain (ind f) D) (fun x y:R => f x y /\ D x)
  (restriction_family f D).

Definition compact (X:R -> Prop) : Prop :=
  forall f:family,
    covering_open_set X f ->
    exists D : R -> Prop, covering_finite X (subfamily f D).

(**********)
Lemma family_P1 :
  forall (f:family) (D:R -> Prop),
    family_open_set f -> family_open_set (subfamily f D).
forall (f0 : family) (D : R -> Prop),
family_open_set f0 -> family_open_set (subfamily f0 D)
Proof.
forall (f0 : family) (D : R -> Prop),
family_open_set f0 -> family_open_set (subfamily f0 D)
  unfold family_open_set; intros; unfold subfamily;
    simpl; assert (H0 := classic (D x)).
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
open_set (fun y : R => f0 x y /\ D x)
  elim H0; intro.
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:D x
open_set (fun y : R => f0 x y /\ D x)
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
open_set (fun y : R => f0 x y /\ D x)
  cut (open_set (f0 x) -> open_set (fun y:R => f0 x y /\ D x)).
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:D x
(open_set (f0 x) ->
 open_set (fun y : R => f0 x y /\ D x)) ->
open_set (fun y : R => f0 x y /\ D x)
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:D x
open_set (f0 x) ->
open_set (fun y : R => f0 x y /\ D x)
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
open_set (fun y : R => f0 x y /\ D x)
  intro; apply H2; apply H.
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:D x
open_set (f0 x) ->
open_set (fun y : R => f0 x y /\ D x)
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
open_set (fun y : R => f0 x y /\ D x)
  unfold open_set; unfold neighbourhood; intros; elim H3;
    intros; assert (H6 := H2 _ H4); elim H6; intros; exists x1;
      unfold included; intros; split.
f0:family
D:R -> Prop
H:forall x3 : R, open_set (f0 x3)
x:R
H0:D x \/ ~ D x
H1:D x
H2:forall x3 : R,
f0 x x3 ->
exists delta : posreal,
  included (disc x3 delta) (f0 x)
x0:R
H3:f0 x x0 /\ D x
H4:f0 x x0
H5:D x
H6:exists delta : posreal,
  included (disc x0 delta) (f0 x)
x1:posreal
H7:included (disc x0 x1) (f0 x)
x2:R
H8:disc x0 x1 x2
f0 x x2
f0:family
D:R -> Prop
H:forall x3 : R, open_set (f0 x3)
x:R
H0:D x \/ ~ D x
H1:D x
H2:forall x3 : R,
f0 x x3 ->
exists delta : posreal,
  included (disc x3 delta) (f0 x)
x0:R
H3:f0 x x0 /\ D x
H4:f0 x x0
H5:D x
H6:exists delta : posreal,
  included (disc x0 delta) (f0 x)
x1:posreal
H7:included (disc x0 x1) (f0 x)
x2:R
H8:disc x0 x1 x2
D x
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
open_set (fun y : R => f0 x y /\ D x)
  apply (H7 _ H8).
f0:family
D:R -> Prop
H:forall x3 : R, open_set (f0 x3)
x:R
H0:D x \/ ~ D x
H1:D x
H2:forall x3 : R,
f0 x x3 ->
exists delta : posreal,
  included (disc x3 delta) (f0 x)
x0:R
H3:f0 x x0 /\ D x
H4:f0 x x0
H5:D x
H6:exists delta : posreal,
  included (disc x0 delta) (f0 x)
x1:posreal
H7:included (disc x0 x1) (f0 x)
x2:R
H8:disc x0 x1 x2
D x
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
open_set (fun y : R => f0 x y /\ D x)
  assumption.
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
open_set (fun y : R => f0 x y /\ D x)
  cut (open_set (fun y:R => False) -> open_set (fun y:R => f0 x y /\ D x)).
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
(open_set (fun _ : R => False) ->
 open_set (fun y : R => f0 x y /\ D x)) ->
open_set (fun y : R => f0 x y /\ D x)
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
open_set (fun _ : R => False) ->
open_set (fun y : R => f0 x y /\ D x)
  intro; apply H2; apply open_set_P4.
f0:family
D:R -> Prop
H:forall x0 : R, open_set (f0 x0)
x:R
H0:D x \/ ~ D x
H1:~ D x
open_set (fun _ : R => False) ->
open_set (fun y : R => f0 x y /\ D x)
  unfold open_set; unfold neighbourhood; intros; elim H3;
    intros; elim H1; assumption.
Qed.

Definition bounded (D:R -> Prop) : Prop :=
  exists m : R, (exists M : R, (forall x:R, D x -> m <= x <= M)).

Lemma open_set_P6 :
  forall D1 D2:R -> Prop, open_set D1 -> D1 =_D D2 -> open_set D2.
forall D1 D2 : R -> Prop,
open_set D1 -> D1 =_D D2 -> open_set D2
Proof.
forall D1 D2 : R -> Prop,
open_set D1 -> D1 =_D D2 -> open_set D2
  unfold open_set; unfold neighbourhood; intros.
D1, D2:R -> Prop
H:forall x0 : R,
D1 x0 ->
exists delta : posreal,
  included (disc x0 delta) D1
H0:D1 =_D D2
x:R
H1:D2 x
exists delta : posreal, included (disc x delta) D2
  unfold eq_Dom in H0; elim H0; intros.
D1, D2:R -> Prop
H:forall x0 : R,
D1 x0 ->
exists delta : posreal,
  included (disc x0 delta) D1
H0:included D1 D2 /\ included D2 D1
x:R
H1:D2 x
H2:included D1 D2
H3:included D2 D1
exists delta : posreal, included (disc x delta) D2
  assert (H4 := H _ (H3 _ H1)).
D1, D2:R -> Prop
H:forall x0 : R,
D1 x0 ->
exists delta : posreal,
  included (disc x0 delta) D1
H0:included D1 D2 /\ included D2 D1
x:R
H1:D2 x
H2:included D1 D2
H3:included D2 D1
H4:exists delta : posreal,
  included (disc x delta) D1
exists delta : posreal, included (disc x delta) D2
  elim H4; intros.
D1, D2:R -> Prop
H:forall x1 : R,
D1 x1 ->
exists delta : posreal,
  included (disc x1 delta) D1
H0:included D1 D2 /\ included D2 D1
x:R
H1:D2 x
H2:included D1 D2
H3:included D2 D1
H4:exists delta : posreal,
  included (disc x delta) D1
x0:posreal
H5:included (disc x x0) D1
exists delta : posreal, included (disc x delta) D2
  exists x0; apply included_trans with D1; assumption.
Qed.

(**********)
Lemma compact_P1 : forall X:R -> Prop, compact X -> bounded X.
forall X : R -> Prop, compact X -> bounded X
Proof.
forall X : R -> Prop, compact X -> bounded X
  intros; unfold compact in H; set (D := fun x:R => True);
    set (g := fun x y:R => Rabs y < x);
      cut (forall x:R, (exists y : _, g x y) -> True);
        [ intro | intro; trivial ].
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
bounded X
  set (f0 := mkfamily D g H0); assert (H1 := H f0);
    cut (covering_open_set X f0).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0 -> bounded X
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  intro; assert (H3 := H1 H2); elim H3; intros D' H4;
    unfold covering_finite in H4; elim H4; intros; unfold family_finite in H6;
      unfold domain_finite in H6; elim H6; intros l H7;
        unfold bounded; set (r := MaxRlist l).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:covering X (subfamily f0 D')
H6:exists l0 : Rlist,
  forall x : R,
  ind (subfamily f0 D') x <-> In x l0
l:Rlist
H7:forall x : R, ind (subfamily f0 D') x <-> In x l
r:=MaxRlist l:R
exists m M : R, forall x : R, X x -> m <= x <= M
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  exists (- r); exists r; intros.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:covering X (subfamily f0 D')
H6:exists l0 : Rlist,
  forall x0 : R,
  ind (subfamily f0 D') x0 <-> In x0 l0
l:Rlist
H7:forall x0 : R,
ind (subfamily f0 D') x0 <-> In x0 l
r:=MaxRlist l:R
x:R
H8:X x
- r <= x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  unfold covering in H5; assert (H9 := H5 _ H8); elim H9; intros;
    unfold subfamily in H10; simpl in H10; elim H10; intros;
      assert (H13 := H7 x0); simpl in H13; cut (intersection_domain D D' x0).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0 -> - r <= x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  elim H13; clear H13; intros.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
- r <= x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  assert (H16 := H13 H15); unfold g in H11; split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
- r <= x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  cut (x0 <= r).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r -> - r <= x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  intro; cut (Rabs x < r).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
Rabs x < r -> - r <= x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
Rabs x < r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  intro; assert (H19 := Rabs_def2 x r H18); elim H19; intros; left; assumption.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
Rabs x < r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  apply Rlt_le_trans with x0; assumption.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  apply (MaxRlist_P1 l x0 H16).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  cut (x0 <= r).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r -> x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  intro; apply Rle_trans with (Rabs x).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
x <= Rabs x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
Rabs x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  apply RRle_abs.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
Rabs x <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  apply Rle_trans with x0.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
Rabs x <= x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  left; apply H11.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
H17:x0 <= r
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  assumption.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:Rabs x < x0
H12:D' x0
H13:intersection_domain D D' x0 -> In x0 l
H14:In x0 l -> intersection_domain D D' x0
H15:intersection_domain D D' x0
H16:In x0 l
x0 <= r
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  apply (MaxRlist_P1 l x0 H16).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x1 y : R => Rabs y < x1:R -> R -> Prop
H0:forall x1 : R, (exists y : R, g x1 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H2:covering_open_set X f0
H3:exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
D':R -> Prop
H4:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H5:forall x1 : R,
X x1 -> exists y : R, subfamily f0 D' y x1
H6:exists l0 : Rlist,
  forall x1 : R,
  ind (subfamily f0 D') x1 <-> In x1 l0
l:Rlist
H7:forall x1 : R,
ind (subfamily f0 D') x1 <-> In x1 l
r:=MaxRlist l:R
x:R
H8:X x
H9:exists y : R, subfamily f0 D' y x
x0:R
H10:g x0 x /\ D' x0
H11:g x0 x
H12:D' x0
H13:intersection_domain D D' x0 <-> In x0 l
intersection_domain D D' x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  unfold intersection_domain, D; tauto.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
  unfold covering_open_set; split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering X f0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
family_open_set f0
  unfold covering; intros; simpl; exists (Rabs x + 1);
    unfold g; pattern (Rabs x) at 1; rewrite <- Rplus_0_r;
      apply Rplus_lt_compat_l; apply Rlt_0_1.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x y : R => Rabs y < x:R -> R -> Prop
H0:forall x : R, (exists y : R, g x y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
family_open_set f0
  unfold family_open_set; intro; case (Rtotal_order 0 x); intro.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 < x
open_set (f0 x)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
open_set (f0 x)
  apply open_set_P6 with (disc 0 (mkposreal _ H2)).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 < x
open_set (disc 0 {| pos := x; cond_pos := H2 |})
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 < x
disc 0 {| pos := x; cond_pos := H2 |} =_D f0 x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
open_set (f0 x)
  apply disc_P1.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 < x
disc 0 {| pos := x; cond_pos := H2 |} =_D f0 x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
open_set (f0 x)
  unfold eq_Dom; unfold f0; simpl;
    unfold g, disc; split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 < x
included
  (fun y : R =>
   Rabs (y - 0) < {| pos := x; cond_pos := H2 |})
  (fun y : R => Rabs y < x)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 < x
included (fun y : R => Rabs y < x)
  (fun y : R =>
   Rabs (y - 0) < {| pos := x; cond_pos := H2 |})
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
open_set (f0 x)
  unfold included; intros; unfold Rminus in H3; rewrite Ropp_0 in H3;
    rewrite Rplus_0_r in H3; apply H3.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 < x
included (fun y : R => Rabs y < x)
  (fun y : R =>
   Rabs (y - 0) < {| pos := x; cond_pos := H2 |})
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
open_set (f0 x)
  unfold included; intros; unfold Rminus; rewrite Ropp_0;
    rewrite Rplus_0_r; apply H3.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
open_set (f0 x)
  apply open_set_P6 with (fun x:R => False).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
open_set (fun _ : R => False)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
(fun _ : R => False) =_D f0 x
  apply open_set_P4.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
(fun _ : R => False) =_D f0 x
  unfold eq_Dom; split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
included (fun _ : R => False) (f0 x)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
included (f0 x) (fun _ : R => False)
  unfold included; intros; elim H3.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
D:=fun _ : R => True:R -> Prop
g:=fun x0 y : R => Rabs y < x0:R -> R -> Prop
H0:forall x0 : R, (exists y : R, g x0 y) -> True
f0:={| ind := D; f := g; cond_fam := H0 |}:family
H1:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H2:0 = x \/ 0 > x
included (f0 x) (fun _ : R => False)
  unfold included, f0; simpl; unfold g; intros; elim H2;
    intro;
      [ rewrite <- H4 in H3; assert (H5 := Rabs_pos x0);
        elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H3))
        | assert (H6 := Rabs_pos x0); assert (H7 := Rlt_trans _ _ _ H3 H4);
          elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 H7)) ].
Qed.

(**********)
Lemma compact_P2 : forall X:R -> Prop, compact X -> closed_set X.
forall X : R -> Prop, compact X -> closed_set X
Proof.
forall X : R -> Prop, compact X -> closed_set X
  intros; assert (H0 := closed_set_P1 X); elim H0; clear H0; intros _ H0;
    apply H0; clear H0.
X:R -> Prop
H:compact X
X =_D adherence X
  unfold eq_Dom; split.
X:R -> Prop
H:compact X
included X (adherence X)
X:R -> Prop
H:compact X
included (adherence X) X
  apply adherence_P1.
X:R -> Prop
H:compact X
included (adherence X) X
  unfold included; unfold adherence;
    unfold point_adherent; intros; unfold compact in H;
      assert (H1 := classic (X x)); elim H1; clear H1; intro.
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:X x
X x
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
X x
  assumption.
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
X x
  cut (forall y:R, X y -> 0 < Rabs (y - x) / 2).
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
(forall y : R, X y -> 0 < Rabs (y - x) / 2) -> X x
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intro; set (D := X);
    set (g := fun y z:R => Rabs (y - z) < Rabs (y - x) / 2 /\ D y);
      cut (forall x:R, (exists y : _, g x y) -> D x).
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
(forall x0 : R, (exists y : R, g x0 y) -> D x0) -> D x
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intro; set (f0 := mkfamily D g H3); assert (H4 := H f0);
    cut (covering_open_set X f0).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0 -> D x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intro; assert (H6 := H4 H5); elim H6; clear H6; intros D' H6.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering_finite X (subfamily f0 D')
D x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold covering_finite in H6; decompose [and] H6;
    unfold covering, subfamily in H7; simpl in H7;
      unfold family_finite, subfamily in H8; simpl in H8;
        unfold domain_finite in H8; elim H8; clear H8; intros l H8;
          set (alp := MinRlist (AbsList l x)); cut (0 < alp).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp -> D x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intro; assert (H10 := H0 (disc x (mkposreal _ H9)));
    cut (neighbourhood (disc x (mkposreal alp H9)) x).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x -> D x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intro; assert (H12 := H10 H11); elim H12; clear H12; intros y H12;
    unfold intersection_domain in H12; elim H12; clear H12;
      intros; assert (H14 := H7 _ H13); elim H14; clear H14;
        intros y0 H14; elim H14; clear H14; intros; unfold g in H14;
          elim H14; clear H14; intros; unfold disc in H12; simpl in H12;
            cut (alp <= Rabs (y0 - x) / 2).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
alp <= Rabs (y0 - x) / 2 -> D x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
alp <= Rabs (y0 - x) / 2
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intro; assert (H18 := Rlt_le_trans _ _ _ H12 H17);
    cut (Rabs (y0 - x) < Rabs (y0 - x)).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
H17:alp <= Rabs (y0 - x) / 2
H18:Rabs (y - x) < Rabs (y0 - x) / 2
Rabs (y0 - x) < Rabs (y0 - x) -> D x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
H17:alp <= Rabs (y0 - x) / 2
H18:Rabs (y - x) < Rabs (y0 - x) / 2
Rabs (y0 - x) < Rabs (y0 - x)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
alp <= Rabs (y0 - x) / 2
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intro; elim (Rlt_irrefl _ H19).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
H17:alp <= Rabs (y0 - x) / 2
H18:Rabs (y - x) < Rabs (y0 - x) / 2
Rabs (y0 - x) < Rabs (y0 - x)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
alp <= Rabs (y0 - x) / 2
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  apply Rle_lt_trans with (Rabs (y0 - y) + Rabs (y - x)).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
H17:alp <= Rabs (y0 - x) / 2
H18:Rabs (y - x) < Rabs (y0 - x) / 2
Rabs (y0 - x) <= Rabs (y0 - y) + Rabs (y - x)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
H17:alp <= Rabs (y0 - x) / 2
H18:Rabs (y - x) < Rabs (y0 - x) / 2
Rabs (y0 - y) + Rabs (y - x) < Rabs (y0 - x)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
alp <= Rabs (y0 - x) / 2
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  replace (y0 - x) with (y0 - y + (y - x)); [ apply Rabs_triang | ring ].
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
H17:alp <= Rabs (y0 - x) / 2
H18:Rabs (y - x) < Rabs (y0 - x) / 2
Rabs (y0 - y) + Rabs (y - x) < Rabs (y0 - x)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
alp <= Rabs (y0 - x) / 2
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  rewrite (double_var (Rabs (y0 - x))); apply Rplus_lt_compat; assumption.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y1 : R, intersection_domain V X y1
H1:~ X x
H2:forall y1 : R, X y1 -> 0 < Rabs (y1 - x) / 2
D:=X:R -> Prop
g:=fun y1 z : R =>
Rabs (y1 - z) < Rabs (y1 - x) / 2 /\ D y1:R -> R -> Prop
H3:forall x0 : R, (exists y1 : R, g x0 y1) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y1 : R, g y1 x0 /\ D' y1
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y1 : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y1
H11:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x
y:R
H12:Rabs (y - x) < alp
H13:X y
y0:R
H15:D' y0
H14:Rabs (y0 - y) < Rabs (y0 - x) / 2
H16:D y0
alp <= Rabs (y0 - x) / 2
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  apply (MinRlist_P1 (AbsList l x) (Rabs (y0 - x) / 2)); apply AbsList_P1;
    elim (H8 y0); clear H8; intros; apply H8; unfold intersection_domain;
      split; assumption.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
neighbourhood
  (disc x {| pos := alp; cond_pos := H9 |}) x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  assert (H11 := disc_P1 x (mkposreal alp H9)); unfold open_set in H11;
    apply H11.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
H9:0 < alp
H10:neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x ->
exists y : R,
intersection_domain (disc x {| pos := alp; cond_pos := H9 |}) X y
H11:forall x0 : R,
disc x {| pos := alp; cond_pos := H9 |} x0 ->
neighbourhood (disc x {| pos := alp; cond_pos := H9 |}) x0
disc x {| pos := alp; cond_pos := H9 |} x
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold disc; unfold Rminus; rewrite Rplus_opp_r;
    rewrite Rabs_R0; apply H9.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
H5:covering_open_set X f0
D':R -> Prop
H6:covering X (subfamily f0 D') /\
family_finite (subfamily f0 D')
H7:forall x0 : R,
X x0 -> exists y : R, g y x0 /\ D' y
l:Rlist
H8:forall x0 : R,
intersection_domain D D' x0 <-> In x0 l
alp:=MinRlist (AbsList l x):R
0 < alp
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold alp; apply MinRlist_P2; intros;
    assert (H10 := AbsList_P2 _ _ _ H9); elim H10; clear H10;
      intros z H10; elim H10; clear H10; intros; rewrite H11;
        apply H2; elim (H8 z); clear H8; intros; assert (H13 := H12 H10);
          unfold intersection_domain, D in H13; elim H13; clear H13;
            intros; assumption.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering_open_set X f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold covering_open_set; split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
covering X f0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
family_open_set f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold covering; intros; exists x0; simpl; unfold g;
    split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:X x0
Rabs (x0 - x0) < Rabs (x0 - x) / 2
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:X x0
D x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
family_open_set f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
    unfold Rminus in H2; apply (H2 _ H5).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:X x0
D x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
family_open_set f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  apply H5.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x0 : R, (exists y : R, g x0 y) -> D x0
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
family_open_set f0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold family_open_set; intro; simpl; unfold g;
    elim (classic (D x0)); intro.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  apply open_set_P6 with (disc x0 (mkposreal _ (H2 _ H5))).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
open_set
  (disc x0
     {|
     pos := Rabs (x0 - x) / 2;
     cond_pos := H2 x0 H5 |})
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
disc x0
  {| pos := Rabs (x0 - x) / 2; cond_pos := H2 x0 H5 |} =_D
(fun z : R =>
 Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  apply disc_P1.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
disc x0
  {| pos := Rabs (x0 - x) / 2; cond_pos := H2 x0 H5 |} =_D
(fun z : R =>
 Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold eq_Dom; split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
included
  (disc x0
     {|
     pos := Rabs (x0 - x) / 2;
     cond_pos := H2 x0 H5 |})
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
included
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
  (disc x0
     {|
     pos := Rabs (x0 - x) / 2;
     cond_pos := H2 x0 H5 |})
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold included, disc; simpl; intros; split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x2 : R, (exists y : R, g x2 y) -> D x2
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
x1:R
H6:Rabs (x1 - x0) < Rabs (x0 - x) / 2
Rabs (x0 - x1) < Rabs (x0 - x) / 2
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x2 : R, (exists y : R, g x2 y) -> D x2
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
x1:R
H6:Rabs (x1 - x0) < Rabs (x0 - x) / 2
D x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
included
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
  (disc x0
     {|
     pos := Rabs (x0 - x) / 2;
     cond_pos := H2 x0 H5 |})
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  rewrite <- (Rabs_Ropp (x0 - x1)); rewrite Ropp_minus_distr; apply H6.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x2 : R, (exists y : R, g x2 y) -> D x2
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
x1:R
H6:Rabs (x1 - x0) < Rabs (x0 - x) / 2
D x0
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
included
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
  (disc x0
     {|
     pos := Rabs (x0 - x) / 2;
     cond_pos := H2 x0 H5 |})
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  apply H5.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:D x0
included
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
  (disc x0
     {|
     pos := Rabs (x0 - x) / 2;
     cond_pos := H2 x0 H5 |})
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold included, disc; simpl; intros; elim H6; intros;
    rewrite <- (Rabs_Ropp (x1 - x0)); rewrite Ropp_minus_distr;
      apply H7.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  apply open_set_P6 with (fun z:R => False).
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
open_set (fun _ : R => False)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
(fun _ : R => False) =_D
(fun z : R =>
 Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  apply open_set_P4.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
(fun _ : R => False) =_D
(fun z : R =>
 Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold eq_Dom; split.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
included (fun _ : R => False)
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
included
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
  (fun _ : R => False)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold included; intros; elim H6.
X:R -> Prop
H:forall f1 : family,
covering_open_set X f1 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f1 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
H3:forall x1 : R, (exists y : R, g x1 y) -> D x1
f0:={| ind := D; f := g; cond_fam := H3 |}:family
H4:covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x0:R
H5:~ D x0
included
  (fun z : R =>
   Rabs (x0 - z) < Rabs (x0 - x) / 2 /\ D x0)
  (fun _ : R => False)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  unfold included; intros; elim H6; intros; elim H5; assumption.
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D0 : R -> Prop,
  covering_finite X (subfamily f0 D0)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
H2:forall y : R, X y -> 0 < Rabs (y - x) / 2
D:=X:R -> Prop
g:=fun y z : R =>
Rabs (y - z) < Rabs (y - x) / 2 /\ D y:R -> R -> Prop
forall x0 : R, (exists y : R, g x0 y) -> D x0
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intros; elim H3; intros; unfold g in H4; elim H4; clear H4; intros _ H4;
    apply H4.
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y : R, intersection_domain V X y
H1:~ X x
forall y : R, X y -> 0 < Rabs (y - x) / 2
  intros; unfold Rdiv; apply Rmult_lt_0_compat.
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y0 : R, intersection_domain V X y0
H1:~ X x
y:R
H2:X y
0 < Rabs (y - x)
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y0 : R, intersection_domain V X y0
H1:~ X x
y:R
H2:X y
0 < / 2
  apply Rabs_pos_lt; apply Rminus_eq_contra; red; intro;
    rewrite H3 in H2; elim H1; apply H2.
X:R -> Prop
H:forall f0 : family,
covering_open_set X f0 ->
exists D : R -> Prop,
  covering_finite X (subfamily f0 D)
x:R
H0:forall V : R -> Prop,
neighbourhood V x ->
exists y0 : R, intersection_domain V X y0
H1:~ X x
y:R
H2:X y
0 < / 2
  apply Rinv_0_lt_compat; prove_sup0.
Qed.

(**********)
Lemma compact_EMP : compact (fun _:R => False).
compact (fun _ : R => False)
Proof.
compact (fun _ : R => False)
  unfold compact; intros; exists (fun x:R => False);
    unfold covering_finite; split.
f0:family
H:covering_open_set (fun _ : R => False) f0
covering (fun _ : R => False)
  (subfamily f0 (fun _ : R => False))
f0:family
H:covering_open_set (fun _ : R => False) f0
family_finite (subfamily f0 (fun _ : R => False))
  unfold covering; intros; elim H0.
f0:family
H:covering_open_set (fun _ : R => False) f0
family_finite (subfamily f0 (fun _ : R => False))
  unfold family_finite; unfold domain_finite; exists nil; intro.
f0:family
H:covering_open_set (fun _ : R => False) f0
x:R
ind (subfamily f0 (fun _ : R => False)) x <-> In x nil
  split.
f0:family
H:covering_open_set (fun _ : R => False) f0
x:R
ind (subfamily f0 (fun _ : R => False)) x -> In x nil
f0:family
H:covering_open_set (fun _ : R => False) f0
x:R
In x nil -> ind (subfamily f0 (fun _ : R => False)) x
  simpl; unfold intersection_domain; intros; elim H0.
f0:family
H:covering_open_set (fun _ : R => False) f0
x:R
H0:ind f0 x /\ False
ind f0 x -> False -> False
f0:family
H:covering_open_set (fun _ : R => False) f0
x:R
In x nil -> ind (subfamily f0 (fun _ : R => False)) x
  elim H0; clear H0; intros _ H0; elim H0.
f0:family
H:covering_open_set (fun _ : R => False) f0
x:R
In x nil -> ind (subfamily f0 (fun _ : R => False)) x
  simpl; intro; elim H0.
Qed.

Lemma compact_eqDom :
  forall X1 X2:R -> Prop, compact X1 -> X1 =_D X2 -> compact X2.
forall X1 X2 : R -> Prop,
compact X1 -> X1 =_D X2 -> compact X2
Proof.
forall X1 X2 : R -> Prop,
compact X1 -> X1 =_D X2 -> compact X2
  unfold compact; intros; unfold eq_Dom in H0; elim H0; clear H0;
    unfold included; intros; assert (H3 : covering_open_set X1 f0).
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D : R -> Prop,
  covering_finite X1 (subfamily f1 D)
f0:family
H1:covering_open_set X2 f0
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
covering_open_set X1 f0
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D : R -> Prop,
  covering_finite X1 (subfamily f1 D)
f0:family
H1:covering_open_set X2 f0
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H3:covering_open_set X1 f0
exists D : R -> Prop,
  covering_finite X2 (subfamily f0 D)
  unfold covering_open_set; unfold covering_open_set in H1; elim H1;
    clear H1; intros; split.
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D : R -> Prop,
  covering_finite X1 (subfamily f1 D)
f0:family
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H1:covering X2 f0
H3:family_open_set f0
covering X1 f0
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D : R -> Prop,
  covering_finite X1 (subfamily f1 D)
f0:family
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H1:covering X2 f0
H3:family_open_set f0
family_open_set f0
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D : R -> Prop,
  covering_finite X1 (subfamily f1 D)
f0:family
H1:covering_open_set X2 f0
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H3:covering_open_set X1 f0
exists D : R -> Prop,
  covering_finite X2 (subfamily f0 D)
  unfold covering in H1; unfold covering; intros;
    apply (H1 _ (H0 _ H4)).
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D : R -> Prop,
  covering_finite X1 (subfamily f1 D)
f0:family
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H1:covering X2 f0
H3:family_open_set f0
family_open_set f0
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D : R -> Prop,
  covering_finite X1 (subfamily f1 D)
f0:family
H1:covering_open_set X2 f0
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H3:covering_open_set X1 f0
exists D : R -> Prop,
  covering_finite X2 (subfamily f0 D)
  apply H3.
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D : R -> Prop,
  covering_finite X1 (subfamily f1 D)
f0:family
H1:covering_open_set X2 f0
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H3:covering_open_set X1 f0
exists D : R -> Prop,
  covering_finite X2 (subfamily f0 D)
  elim (H _ H3); intros D H4; exists D; unfold covering_finite;
    unfold covering_finite in H4; elim H4; intros; split.
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D0 : R -> Prop,
  covering_finite X1 (subfamily f1 D0)
f0:family
H1:covering_open_set X2 f0
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H3:covering_open_set X1 f0
D:R -> Prop
H4:covering X1 (subfamily f0 D) /\
family_finite (subfamily f0 D)
H5:covering X1 (subfamily f0 D)
H6:family_finite (subfamily f0 D)
covering X2 (subfamily f0 D)
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D0 : R -> Prop,
  covering_finite X1 (subfamily f1 D0)
f0:family
H1:covering_open_set X2 f0
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H3:covering_open_set X1 f0
D:R -> Prop
H4:covering X1 (subfamily f0 D) /\
family_finite (subfamily f0 D)
H5:covering X1 (subfamily f0 D)
H6:family_finite (subfamily f0 D)
family_finite (subfamily f0 D)
  unfold covering in H5; unfold covering; intros;
    apply (H5 _ (H2 _ H7)).
X1, X2:R -> Prop
H:forall f1 : family,
covering_open_set X1 f1 ->
exists D0 : R -> Prop,
  covering_finite X1 (subfamily f1 D0)
f0:family
H1:covering_open_set X2 f0
H0:forall x : R, X1 x -> X2 x
H2:forall x : R, X2 x -> X1 x
H3:covering_open_set X1 f0
D:R -> Prop
H4:covering X1 (subfamily f0 D) /\
family_finite (subfamily f0 D)
H5:covering X1 (subfamily f0 D)
H6:family_finite (subfamily f0 D)
family_finite (subfamily f0 D)
  apply H6.
Qed.