:: On the Subcontinua of a Real Line
:: by Adam Grabowski
::
:: Received June 12, 2003
:: Copyright (c) 2003-2012 Association of Mizar Users


begin

theorem :: BORSUK_5:1
canceled;

theorem :: BORSUK_5:2
for x1, x2, x3, x4, x5, x6 being set holds {x1,x2,x3,x4,x5,x6} = {x1,x3,x6} \/ {x2,x4,x5}
proof end;

theorem Th3: :: BORSUK_5:3
for x1, x2, x3, x4, x5, x6 being set st x1,x2,x3,x4,x5,x6 are_mutually_different holds
card {x1,x2,x3,x4,x5,x6} = 6
proof end;

theorem :: BORSUK_5:4
for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds
card {x1,x2,x3,x4,x5,x6,x7} = 7
proof end;

theorem Th5: :: BORSUK_5:5
for x1, x2, x3, x4, x5, x6 being set st {x1,x2,x3} misses {x4,x5,x6} holds
( x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 )
proof end;

theorem :: BORSUK_5:6
for x1, x2, x3, x4, x5, x6 being set st x1,x2,x3 are_mutually_different & x4,x5,x6 are_mutually_different & {x1,x2,x3} misses {x4,x5,x6} holds
x1,x2,x3,x4,x5,x6 are_mutually_different
proof end;

theorem :: BORSUK_5:7
for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6 are_mutually_different & {x1,x2,x3,x4,x5,x6} misses {x7} holds
x1,x2,x3,x4,x5,x6,x7 are_mutually_different
proof end;

theorem :: BORSUK_5:8
for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds
x7,x1,x2,x3,x4,x5,x6 are_mutually_different
proof end;

theorem :: BORSUK_5:9
for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds
x1,x2,x5,x3,x6,x7,x4 are_mutually_different
proof end;

Lm1: R^1 is pathwise_connected
proof end;

registration
cluster R^1 -> pathwise_connected ;
coherence
R^1 is pathwise_connected by Lm1;
end;

registration
cluster non empty TopSpace-like connected for TopStruct ;
existence
ex b1 being TopSpace st
( b1 is connected & not b1 is empty )
proof end;
end;

begin

theorem :: BORSUK_5:10
for A, B being Subset of R^1
for a, b, c, d being real number st a < b & b <= c & c < d & A = [.a,b.[ & B = ].c,d.] holds
A,B are_separated
proof end;

theorem Th11: :: BORSUK_5:11
for a, b, c being real number st a <= c & c <= b holds
[.a,b.] \/ [.c,+infty.[ = [.a,+infty.[
proof end;

theorem Th12: :: BORSUK_5:12
for a, b, c being real number st a <= c & c <= b holds
].-infty,c.] \/ [.a,b.] = ].-infty,b.]
proof end;

registration
cluster -> real for Element of RAT ;
coherence
for b1 being Element of RAT holds b1 is real ;
end;

theorem :: BORSUK_5:13
for A being Subset of R^1
for p being Point of RealSpace holds
( p in Cl A iff for r being real number st r > 0 holds
Ball (p,r) meets A ) by GOBOARD6:92, TOPMETR:def 6;

theorem Th14: :: BORSUK_5:14
for p, q being Element of RealSpace st q >= p holds
dist (p,q) = q - p
proof end;

theorem Th15: :: BORSUK_5:15
for A being Subset of R^1 st A = RAT holds
Cl A = the carrier of R^1
proof end;

theorem Th16: :: BORSUK_5:16
for A being Subset of R^1
for a, b being real number st A = ].a,b.[ & a <> b holds
Cl A = [.a,b.]
proof end;

begin

registration
cluster number_e -> irrational ;
coherence
number_e is irrational by IRRAT_1:41;
end;

definition
func IRRAT -> Subset of REAL equals :: BORSUK_5:def 1
REAL \ RAT;
coherence
REAL \ RAT is Subset of REAL ;
end;

:: deftheorem defines IRRAT BORSUK_5:def 1 :
IRRAT = REAL \ RAT;

definition
let a, b be real number ;
func RAT (a,b) -> Subset of REAL equals :: BORSUK_5:def 2
RAT /\ ].a,b.[;
coherence
RAT /\ ].a,b.[ is Subset of REAL ;
func IRRAT (a,b) -> Subset of REAL equals :: BORSUK_5:def 3
IRRAT /\ ].a,b.[;
coherence
IRRAT /\ ].a,b.[ is Subset of REAL ;
end;

:: deftheorem defines RAT BORSUK_5:def 2 :
for a, b being real number holds RAT (a,b) = RAT /\ ].a,b.[;

:: deftheorem defines IRRAT BORSUK_5:def 3 :
for a, b being real number holds IRRAT (a,b) = IRRAT /\ ].a,b.[;

theorem Th17: :: BORSUK_5:17
for x being real number holds
( x is irrational iff x in IRRAT )
proof end;

registration
cluster complex real ext-real irrational for set ;
existence
ex b1 being real number st b1 is irrational by IRRAT_1:41;
end;

registration
cluster IRRAT -> non empty ;
coherence
not IRRAT is empty by Th17, IRRAT_1:41;
end;

registration
let a be rational number ;
let b be real irrational number ;
cluster a + b -> irrational ;
coherence
a + b is irrational
proof end;
cluster b + a -> irrational ;
coherence
b + a is irrational ;
end;

theorem :: BORSUK_5:18
for a being rational number
for b being real irrational number holds a + b is irrational ;

registration
let a be real irrational number ;
cluster - a -> irrational ;
coherence
- a is irrational
proof end;
end;

theorem :: BORSUK_5:19
for a being real irrational number holds - a is irrational ;

registration
let a be rational number ;
let b be real irrational number ;
cluster a - b -> irrational ;
coherence
a - b is irrational
proof end;
cluster b - a -> irrational ;
coherence
b - a is irrational
proof end;
end;

theorem :: BORSUK_5:20
for a being rational number
for b being real irrational number holds a - b is irrational ;

theorem :: BORSUK_5:21
for a being rational number
for b being real irrational number holds b - a is irrational ;

theorem Th22: :: BORSUK_5:22
for a being rational number
for b being real irrational number st a <> 0 holds
a * b is irrational
proof end;

theorem Th23: :: BORSUK_5:23
for a being rational number
for b being real irrational number st a <> 0 holds
b / a is irrational
proof end;

registration
cluster real irrational -> non zero real for set ;
coherence
for b1 being real number st b1 is irrational holds
not b1 is zero
proof end;
end;

theorem Th24: :: BORSUK_5:24
for a being rational number
for b being real irrational number st a <> 0 holds
a / b is irrational
proof end;

registration
let r be real irrational number ;
cluster frac r -> irrational ;
coherence
frac r is irrational
proof end;
end;

theorem :: BORSUK_5:25
for r being real irrational number holds frac r is irrational ;

registration
cluster non zero complex real ext-real rational for set ;
existence
not for b1 being rational number holds b1 is zero
proof end;
end;

registration
let a be non zero rational number ;
let b be real irrational number ;
cluster a * b -> irrational ;
coherence
a * b is irrational by Th22;
cluster b * a -> irrational ;
coherence
b * a is irrational ;
cluster a / b -> irrational ;
coherence
a / b is irrational by Th24;
cluster b / a -> irrational ;
coherence
b / a is irrational by Th23;
end;

theorem Th26: :: BORSUK_5:26
for a, b being real number st a < b holds
ex p1, p2 being rational number st
( a < p1 & p1 < p2 & p2 < b )
proof end;

theorem Th27: :: BORSUK_5:27
for a, b being real number st a < b holds
ex p being real irrational number st
( a < p & p < b )
proof end;

theorem Th28: :: BORSUK_5:28
for A being Subset of R^1 st A = IRRAT holds
Cl A = the carrier of R^1
proof end;

Lm2: for A being Subset of R^1
for a, b being real number st a < b & A = RAT (a,b) holds
( a in Cl A & b in Cl A )
proof end;

Lm3: for A being Subset of R^1
for a, b being real number st a < b & A = IRRAT (a,b) holds
( a in Cl A & b in Cl A )
proof end;

theorem Th29: :: BORSUK_5:29
for a, b, c being real number holds
( c in RAT (a,b) iff ( c is rational & a < c & c < b ) )
proof end;

theorem Th30: :: BORSUK_5:30
for a, b, c being real number holds
( c in IRRAT (a,b) iff ( c is irrational & a < c & c < b ) )
proof end;

theorem Th31: :: BORSUK_5:31
for A being Subset of R^1
for a, b being real number st a < b & A = RAT (a,b) holds
Cl A = [.a,b.]
proof end;

theorem Th32: :: BORSUK_5:32
for A being Subset of R^1
for a, b being real number st a < b & A = IRRAT (a,b) holds
Cl A = [.a,b.]
proof end;

theorem Th33: :: BORSUK_5:33
for T being connected TopSpace
for A being open closed Subset of T holds
( A = {} or A = [#] T )
proof end;

theorem Th34: :: BORSUK_5:34
for A being Subset of R^1 st A is closed & A is open & not A = {} holds
A = REAL
proof end;

begin

theorem Th35: :: BORSUK_5:35
for A being Subset of R^1
for a, b being real number st A = [.a,b.[ & a <> b holds
Cl A = [.a,b.]
proof end;

theorem Th36: :: BORSUK_5:36
for A being Subset of R^1
for a, b being real number st A = ].a,b.] & a <> b holds
Cl A = [.a,b.]
proof end;

theorem :: BORSUK_5:37
for A being Subset of R^1
for a, b, c being real number st A = [.a,b.[ \/ ].b,c.] & a < b & b < c holds
Cl A = [.a,c.]
proof end;

theorem Th38: :: BORSUK_5:38
for A being Subset of R^1
for a being real number st A = {a} holds
Cl A = {a}
proof end;

theorem Th39: :: BORSUK_5:39
for A being Subset of REAL
for B being Subset of R^1 st A = B holds
( A is open iff B is open )
proof end;

Lm4: for a being real number holds ].-infty,a.] is closed
proof end;

Lm5: for a being real number holds [.a,+infty.[ is closed
proof end;

registration
let A, B be open Subset of REAL;
reconsider A1 = A, B1 = B as Subset of R^1 by TOPMETR:17;
cluster A /\ B -> open for Subset of REAL;
coherence
for b1 being Subset of REAL st b1 = A /\ B holds
b1 is open
proof end;
cluster A \/ B -> open for Subset of REAL;
coherence
for b1 being Subset of REAL st b1 = A \/ B holds
b1 is open
proof end;
end;

Lm6: for a, b being ext-real number holds ].a,b.[ is open
proof end;

theorem Th40: :: BORSUK_5:40
for A being Subset of R^1
for a, b being ext-real number st A = ].a,b.[ holds
A is open
proof end;

theorem Th41: :: BORSUK_5:41
for A being Subset of R^1
for a being real number st A = ].-infty,a.] holds
A is closed
proof end;

theorem Th42: :: BORSUK_5:42
for A being Subset of R^1
for a being real number st A = [.a,+infty.[ holds
A is closed
proof end;

theorem Th43: :: BORSUK_5:43
for a being real number holds [.a,+infty.[ = {a} \/ ].a,+infty.[
proof end;

theorem Th44: :: BORSUK_5:44
for a being real number holds ].-infty,a.] = {a} \/ ].-infty,a.[
proof end;

registration
let a be real number ;
cluster K451(a,+infty) -> non empty ;
coherence
not ].a,+infty.[ is empty
proof end;
cluster K450(-infty,a) -> non empty ;
coherence
not ].-infty,a.] is empty by XXREAL_1:234;
cluster K451(-infty,a) -> non empty ;
coherence
not ].-infty,a.[ is empty
proof end;
cluster K449(a,+infty) -> non empty ;
coherence
not [.a,+infty.[ is empty by XXREAL_1:236;
end;

theorem Th45: :: BORSUK_5:45
for a being real number holds ].a,+infty.[ <> REAL
proof end;

theorem :: BORSUK_5:46
for a being real number holds [.a,+infty.[ <> REAL
proof end;

theorem :: BORSUK_5:47
for a being real number holds ].-infty,a.] <> REAL
proof end;

theorem Th48: :: BORSUK_5:48
for a being real number holds ].-infty,a.[ <> REAL
proof end;

theorem Th49: :: BORSUK_5:49
for A being Subset of R^1
for a being real number st A = ].a,+infty.[ holds
Cl A = [.a,+infty.[
proof end;

theorem :: BORSUK_5:50
for a being real number holds Cl ].a,+infty.[ = [.a,+infty.[
proof end;

theorem Th51: :: BORSUK_5:51
for A being Subset of R^1
for a being real number st A = ].-infty,a.[ holds
Cl A = ].-infty,a.]
proof end;

theorem :: BORSUK_5:52
for a being real number holds Cl ].-infty,a.[ = ].-infty,a.]
proof end;

theorem Th53: :: BORSUK_5:53
for A, B being Subset of R^1
for b being real number st A = ].-infty,b.[ & B = ].b,+infty.[ holds
A,B are_separated
proof end;

theorem :: BORSUK_5:54
for A being Subset of R^1
for a, b being real number st a < b & A = [.a,b.[ \/ ].b,+infty.[ holds
Cl A = [.a,+infty.[
proof end;

theorem Th55: :: BORSUK_5:55
for A being Subset of R^1
for a, b being real number st a < b & A = ].a,b.[ \/ ].b,+infty.[ holds
Cl A = [.a,+infty.[
proof end;

theorem :: BORSUK_5:56
for A being Subset of R^1
for a, b, c being real number st a < b & b < c & A = ((RAT (a,b)) \/ ].b,c.[) \/ ].c,+infty.[ holds
Cl A = [.a,+infty.[
proof end;

theorem Th57: :: BORSUK_5:57
for a, b being real number holds IRRAT (a,b) misses RAT (a,b)
proof end;

theorem Th58: :: BORSUK_5:58
for a, b being real number holds REAL \ (RAT (a,b)) = (].-infty,a.] \/ (IRRAT (a,b))) \/ [.b,+infty.[
proof end;

theorem Th59: :: BORSUK_5:59
for a, b being real number st a < b holds
[.a,+infty.[ \ (].a,b.[ \/ ].b,+infty.[) = {a} \/ {b}
proof end;

theorem :: BORSUK_5:60
for A being Subset of R^1 st A = ((RAT (2,4)) \/ ].4,5.[) \/ ].5,+infty.[ holds
A ` = ((].-infty,2.] \/ (IRRAT (2,4))) \/ {4}) \/ {5}
proof end;

theorem :: BORSUK_5:61
for A being Subset of R^1
for a being real number st A = {a} holds
A ` = ].-infty,a.[ \/ ].a,+infty.[ by TOPMETR:17, XXREAL_1:389;

Lm7: ((IRRAT (2,4)) \/ {4}) \/ {5} c= ].1,+infty.[
proof end;

Lm8: ].1,+infty.[ c= [.1,+infty.[
by XXREAL_1:45;

Lm9: ].-infty,1.[ /\ (((].-infty,2.] \/ (IRRAT (2,4))) \/ {4}) \/ {5}) = ].-infty,1.[
proof end;

Lm10: ].1,+infty.[ /\ (((].-infty,2.] \/ (IRRAT (2,4))) \/ {4}) \/ {5}) = ((].1,2.] \/ (IRRAT (2,4))) \/ {4}) \/ {5}
proof end;

theorem :: BORSUK_5:62
(].-infty,1.[ \/ ].1,+infty.[) /\ (((].-infty,2.] \/ (IRRAT (2,4))) \/ {4}) \/ {5}) = (((].-infty,1.[ \/ ].1,2.]) \/ (IRRAT (2,4))) \/ {4}) \/ {5}
proof end;

theorem :: BORSUK_5:63
for A being Subset of R^1
for a, b being real number st a <= b & A = {a} \/ [.b,+infty.[ holds
A ` = ].-infty,a.[ \/ ].a,b.[
proof end;

theorem :: BORSUK_5:64
for A being Subset of R^1
for a, b being real number st a < b & A = ].-infty,a.[ \/ ].a,b.[ holds
Cl A = ].-infty,b.]
proof end;

theorem Th65: :: BORSUK_5:65
for A being Subset of R^1
for a, b being real number st a < b & A = ].-infty,a.[ \/ ].a,b.] holds
Cl A = ].-infty,b.]
proof end;

theorem Th66: :: BORSUK_5:66
for A being Subset of R^1
for a, b, c being real number st a < b & b < c & A = ((].-infty,a.[ \/ ].a,b.]) \/ (IRRAT (b,c))) \/ {c} holds
Cl A = ].-infty,c.]
proof end;

theorem :: BORSUK_5:67
for A being Subset of R^1
for a, b, c, d being real number st a < b & b < c & A = (((].-infty,a.[ \/ ].a,b.]) \/ (IRRAT (b,c))) \/ {c}) \/ {d} holds
Cl A = ].-infty,c.] \/ {d}
proof end;

theorem :: BORSUK_5:68
for A being Subset of R^1
for a, b being real number st a <= b & A = ].-infty,a.] \/ {b} holds
A ` = ].a,b.[ \/ ].b,+infty.[
proof end;

theorem :: BORSUK_5:69
for a, b being real number holds [.a,+infty.[ \/ {b} <> REAL
proof end;

theorem :: BORSUK_5:70
for a, b being real number holds ].-infty,a.] \/ {b} <> REAL
proof end;

theorem :: BORSUK_5:71
for TS being TopStruct
for A, B being Subset of TS st A <> B holds
A ` <> B `
proof end;

theorem :: BORSUK_5:72
for A being Subset of R^1 st REAL = A ` holds
A = {}
proof end;

begin

theorem Th73: :: BORSUK_5:73
for X being compact Subset of R^1
for X9 being Subset of REAL st X9 = X holds
( X9 is bounded_above & X9 is bounded_below )
proof end;

theorem Th74: :: BORSUK_5:74
for X being compact Subset of R^1
for X9 being Subset of REAL
for x being real number st x in X9 & X9 = X holds
( lower_bound X9 <= x & x <= upper_bound X9 )
proof end;

theorem Th75: :: BORSUK_5:75
for C being non empty connected compact Subset of R^1
for C9 being Subset of REAL st C = C9 & [.(lower_bound C9),(upper_bound C9).] c= C9 holds
[.(lower_bound C9),(upper_bound C9).] = C9
proof end;

theorem Th76: :: BORSUK_5:76
for A being connected Subset of R^1
for a, b, c being real number st a <= b & b <= c & a in A & c in A holds
b in A
proof end;

theorem Th77: :: BORSUK_5:77
for A being connected Subset of R^1
for a, b being real number st a in A & b in A holds
[.a,b.] c= A
proof end;

theorem Th78: :: BORSUK_5:78
for X being non empty connected compact Subset of R^1 holds X is non empty closed_interval Subset of REAL
proof end;

theorem :: BORSUK_5:79
for A being non empty connected compact Subset of R^1 ex a, b being real number st
( a <= b & A = [.a,b.] )
proof end;

registration
let T be TopSpace;
cluster non empty open closed for Element of bool (bool the carrier of T);
existence
ex b1 being Subset-Family of T st
( b1 is open & b1 is closed & not b1 is empty )
proof end;
end;