:: On the Subcontinua of a Real Line
:: by Adam Grabowski
::
:: Received June 12, 2003
:: Copyright (c) 2003 Association of Mizar Users
theorem :: BORSUK_5:1
canceled;
theorem Th2: :: BORSUK_5:2
theorem :: BORSUK_5:3
for
x1,
x2,
x3,
x4,
x5,
x6 being
set holds
{x1,x2,x3,x4,x5,x6} = {x1,x3,x6} \/ {x2,x4,x5}
definition
canceled;
canceled;
end;
:: deftheorem BORSUK_5:def 1 :
canceled;
:: deftheorem BORSUK_5:def 2 :
canceled;
theorem Th4: :: BORSUK_5:4
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
theorem :: BORSUK_5:5
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
theorem Th6: :: BORSUK_5:6
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 )
theorem :: BORSUK_5:7
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
theorem :: BORSUK_5:8
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
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
x7,
x1,
x2,
x3,
x4,
x5,
x6 are_mutually_different
theorem :: BORSUK_5:10
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
Lm1:
R^1 is arcwise_connected
theorem :: BORSUK_5:11
canceled;
theorem :: BORSUK_5:12
canceled;
theorem :: BORSUK_5:13
canceled;
theorem :: BORSUK_5:14
canceled;
theorem :: BORSUK_5:15
canceled;
theorem :: BORSUK_5:16
canceled;
theorem :: BORSUK_5:17
canceled;
theorem :: BORSUK_5:18
canceled;
theorem :: BORSUK_5:19
canceled;
theorem :: BORSUK_5:20
canceled;
theorem :: BORSUK_5:21
theorem :: BORSUK_5:22
canceled;
theorem :: BORSUK_5:23
canceled;
theorem :: BORSUK_5:24
canceled;
theorem :: BORSUK_5:25
canceled;
theorem :: BORSUK_5:26
canceled;
theorem :: BORSUK_5:27
canceled;
theorem Th28: :: BORSUK_5:28
theorem Th29: :: BORSUK_5:29
theorem :: BORSUK_5:30
canceled;
theorem :: BORSUK_5:31
canceled;
theorem :: BORSUK_5:32
canceled;
theorem :: BORSUK_5:33
theorem Th34: :: BORSUK_5:34
theorem Th35: :: BORSUK_5:35
theorem Th36: :: BORSUK_5:36
:: deftheorem defines IRRAT BORSUK_5:def 3 :
:: deftheorem defines RAT BORSUK_5:def 4 :
:: deftheorem defines IRRAT BORSUK_5:def 5 :
theorem Th37: :: BORSUK_5:37
theorem Th38: :: BORSUK_5:38
theorem Th39: :: BORSUK_5:39
theorem Th40: :: BORSUK_5:40
theorem Th41: :: BORSUK_5:41
theorem Th42: :: BORSUK_5:42
theorem Th43: :: BORSUK_5:43
theorem Th44: :: BORSUK_5:44
theorem Th45: :: BORSUK_5:45
theorem :: BORSUK_5:46
canceled;
theorem Th47: :: BORSUK_5:47
theorem :: BORSUK_5:48
canceled;
theorem :: BORSUK_5:49
canceled;
theorem Th50: :: BORSUK_5:50
theorem Th51: :: BORSUK_5:51
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 )
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 )
theorem Th52: :: BORSUK_5:52
theorem Th53: :: BORSUK_5:53
theorem Th54: :: BORSUK_5:54
theorem Th55: :: BORSUK_5:55
theorem Th56: :: BORSUK_5:56
theorem Th57: :: BORSUK_5:57
theorem Th58: :: BORSUK_5:58
theorem Th59: :: BORSUK_5:59
theorem :: BORSUK_5:60
theorem Th61: :: BORSUK_5:61
theorem Th62: :: BORSUK_5:62
Lm4:
for a being real number holds ].-infty ,a.] is closed
Lm5:
for a being real number holds [.a,+infty .[ is closed
Lm6:
for a, b being ext-real number holds ].a,b.[ is open
theorem Th63: :: BORSUK_5:63
theorem :: BORSUK_5:64
canceled;
theorem Th65: :: BORSUK_5:65
theorem Th66: :: BORSUK_5:66
theorem Th67: :: BORSUK_5:67
theorem Th68: :: BORSUK_5:68
theorem :: BORSUK_5:69
canceled;
theorem :: BORSUK_5:70
canceled;
theorem Th71: :: BORSUK_5:71
theorem :: BORSUK_5:72
theorem :: BORSUK_5:73
theorem Th74: :: BORSUK_5:74
theorem Th75: :: BORSUK_5:75
theorem :: BORSUK_5:76
theorem Th77: :: BORSUK_5:77
theorem :: BORSUK_5:78
theorem Th79: :: BORSUK_5:79
theorem :: BORSUK_5:80
theorem Th81: :: BORSUK_5:81
theorem :: BORSUK_5:82
theorem :: BORSUK_5:83
canceled;
theorem Th84: :: BORSUK_5:84
Lm7:
for a, b being real number st b <= a holds
RAT a,b = {}
Lm8:
for a, b being real number st b <= a holds
REAL = ].-infty ,a.] \/ [.b,+infty .[
theorem Th85: :: BORSUK_5:85
theorem :: BORSUK_5:86
canceled;
theorem :: BORSUK_5:87
canceled;
theorem Th88: :: BORSUK_5:88
theorem :: BORSUK_5:89
theorem :: BORSUK_5:90
Lm9:
((IRRAT 2,4) \/ {4}) \/ {5} c= ].1,+infty .[
Lm10:
].1,+infty .[ c= [.1,+infty .[
by XXREAL_1:45;
Lm11:
].-infty ,1.[ /\ (((].-infty ,2.] \/ (IRRAT 2,4)) \/ {4}) \/ {5}) = ].-infty ,1.[
Lm12:
].1,+infty .[ /\ (((].-infty ,2.] \/ (IRRAT 2,4)) \/ {4}) \/ {5}) = ((].1,2.] \/ (IRRAT 2,4)) \/ {4}) \/ {5}
theorem :: BORSUK_5:91
canceled;
theorem :: BORSUK_5:92
theorem :: BORSUK_5:93
canceled;
theorem :: BORSUK_5:94
canceled;
theorem :: BORSUK_5:95
theorem :: BORSUK_5:96
theorem Th97: :: BORSUK_5:97
theorem :: BORSUK_5:98
canceled;
theorem :: BORSUK_5:99
canceled;
theorem Th100: :: BORSUK_5:100
theorem :: BORSUK_5:101
theorem :: BORSUK_5:102
theorem :: BORSUK_5:103
theorem :: BORSUK_5:104
theorem :: BORSUK_5:105
theorem :: BORSUK_5:106
theorem Th107: :: BORSUK_5:107
theorem Th108: :: BORSUK_5:108
theorem Th109: :: BORSUK_5:109
theorem Th110: :: BORSUK_5:110
theorem Th111: :: BORSUK_5:111
theorem Th112: :: BORSUK_5:112
theorem :: BORSUK_5:113