:: JORDAN2C semantic presentation
theorem Th1: :: JORDAN2C:1
for b
1 being
Real holds
( b
1 <= 0 implies
abs b
1 = - b
1 )
theorem Th2: :: JORDAN2C:2
for b
1, b
2 being
Nat holds
not ( b
1 <= b
2 & b
2 <= b
1 + 2 & not b
2 = b
1 & not b
2 = b
1 + 1 & not b
2 = b
1 + 2 )
theorem Th3: :: JORDAN2C:3
for b
1, b
2 being
Nat holds
not ( b
1 <= b
2 & b
2 <= b
1 + 3 & not b
2 = b
1 & not b
2 = b
1 + 1 & not b
2 = b
1 + 2 & not b
2 = b
1 + 3 )
theorem Th4: :: JORDAN2C:4
for b
1, b
2 being
Nat holds
not ( b
1 <= b
2 & b
2 <= b
1 + 4 & not b
2 = b
1 & not b
2 = b
1 + 1 & not b
2 = b
1 + 2 & not b
2 = b
1 + 3 & not b
2 = b
1 + 4 )
Lemma5:
for b1, b2 being real number holds
( b1 >= 0 & b2 >= 0 implies b1 + b2 >= 0 )
by XREAL_1:35;
Lemma6:
for b1, b2 being real number holds
not ( b1 > 0 & b2 >= 0 & not b1 + b2 > 0 )
by XREAL_1:36;
theorem Th5: :: JORDAN2C:5
canceled;
theorem Th6: :: JORDAN2C:6
canceled;
theorem Th7: :: JORDAN2C:7
for b
1, b
2 being
set for b
3 being
FinSequence holds
not (
rng b
3 = {b1,b2} &
len b
3 = 2 & not ( b
3 . 1
= b
1 & b
3 . 2
= b
2 ) & not ( b
3 . 1
= b
2 & b
3 . 2
= b
1 ) )
theorem Th8: :: JORDAN2C:8
theorem Th9: :: JORDAN2C:9
theorem Th10: :: JORDAN2C:10
theorem Th11: :: JORDAN2C:11
theorem Th12: :: JORDAN2C:12
theorem Th13: :: JORDAN2C:13
theorem Th14: :: JORDAN2C:14
theorem Th15: :: JORDAN2C:15
:: deftheorem Def1 JORDAN2C:def 1 :
canceled;
:: deftheorem Def2 defines Bounded JORDAN2C:def 2 :
theorem Th16: :: JORDAN2C:16
:: deftheorem Def3 defines is_inside_component_of JORDAN2C:def 3 :
theorem Th17: :: JORDAN2C:17
:: deftheorem Def4 defines is_outside_component_of JORDAN2C:def 4 :
theorem Th18: :: JORDAN2C:18
theorem Th19: :: JORDAN2C:19
theorem Th20: :: JORDAN2C:20
:: deftheorem Def5 defines BDD JORDAN2C:def 5 :
:: deftheorem Def6 defines UBD JORDAN2C:def 6 :
theorem Th21: :: JORDAN2C:21
theorem Th22: :: JORDAN2C:22
theorem Th23: :: JORDAN2C:23
theorem Th24: :: JORDAN2C:24
theorem Th25: :: JORDAN2C:25
theorem Th26: :: JORDAN2C:26
theorem Th27: :: JORDAN2C:27
theorem Th28: :: JORDAN2C:28
theorem Th29: :: JORDAN2C:29
theorem Th30: :: JORDAN2C:30
theorem Th31: :: JORDAN2C:31
theorem Th32: :: JORDAN2C:32
canceled;
Lemma32:
for b1 being non empty TopSpace
for b2, b3, b4 being Point of b1
for b5, b6 being Function of I[01] ,b1 holds
not ( b5 is continuous & b2 = b5 . 0 & b3 = b5 . 1 & b6 is continuous & b3 = b6 . 0 & b4 = b6 . 1 & ( for b7 being Function of I[01] ,b1 holds
not ( b7 is continuous & b2 = b7 . 0 & b4 = b7 . 1 & rng b7 c= (rng b5) \/ (rng b6) ) ) )
by BORSUK_2:16;
theorem Th33: :: JORDAN2C:33
:: deftheorem Def7 defines 1* JORDAN2C:def 7 :
:: deftheorem Def8 defines 1.REAL JORDAN2C:def 8 :
theorem Th34: :: JORDAN2C:34
theorem Th35: :: JORDAN2C:35
theorem Th36: :: JORDAN2C:36
theorem Th37: :: JORDAN2C:37
theorem Th38: :: JORDAN2C:38
theorem Th39: :: JORDAN2C:39
theorem Th40: :: JORDAN2C:40
theorem Th41: :: JORDAN2C:41
theorem Th42: :: JORDAN2C:42
theorem Th43: :: JORDAN2C:43
theorem Th44: :: JORDAN2C:44
theorem Th45: :: JORDAN2C:45
theorem Th46: :: JORDAN2C:46
for b
1 being
Natfor b
2 being
Subset of
(TOP-REAL b1)for b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Point of
(TOP-REAL b1) holds
not ( b
3 in b
2 & b
4 in b
2 & b
5 in b
2 & b
6 in b
2 & b
7 in b
2 & b
8 in b
2 & b
9 in b
2 &
LSeg b
3,b
4 c= b
2 &
LSeg b
4,b
5 c= b
2 &
LSeg b
5,b
6 c= b
2 &
LSeg b
6,b
7 c= b
2 &
LSeg b
7,b
8 c= b
2 &
LSeg b
8,b
9 c= b
2 & ( for b
10 being
Function of
I[01] ,
((TOP-REAL b1) | b2) holds
not ( b
10 is
continuous & b
3 = b
10 . 0 & b
9 = b
10 . 1 ) ) )
theorem Th47: :: JORDAN2C:47
theorem Th48: :: JORDAN2C:48
theorem Th49: :: JORDAN2C:49
theorem Th50: :: JORDAN2C:50
theorem Th51: :: JORDAN2C:51
canceled;
theorem Th52: :: JORDAN2C:52
theorem Th53: :: JORDAN2C:53
theorem Th54: :: JORDAN2C:54
theorem Th55: :: JORDAN2C:55
for b
1 being
Natfor b
2 being
Element of
REAL b
1 holds
not ( b
1 >= 2 & b
2 <> 0* b
1 & ( for b
3 being
Element of
REAL b
1 holds
not for b
4 being
Real holds
( not b
3 = b
4 * b
2 & not b
2 = b
4 * b
3 ) ) )
theorem Th56: :: JORDAN2C:56
for b
1 being
Natfor b
2 being
Realfor b
3 being
Subset of
(TOP-REAL b1)for b
4, b
5 being
Point of
(TOP-REAL b1) holds
not ( b
1 >= 2 & b
3 = { b6 where B is Point of (TOP-REAL b1) : |.b6.| > b2 } & b
4 in b
3 & b
5 in b
3 & not for b
6 being
Real holds
( not b
4 = b
6 * b
5 & not b
5 = b
6 * b
4 ) & ( for b
6, b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL b1) holds
not ( b
6 in b
3 & b
7 in b
3 & b
8 in b
3 & b
9 in b
3 & b
10 in b
3 &
LSeg b
4,b
6 c= b
3 &
LSeg b
6,b
7 c= b
3 &
LSeg b
7,b
8 c= b
3 &
LSeg b
8,b
9 c= b
3 &
LSeg b
9,b
10 c= b
3 &
LSeg b
10,b
5 c= b
3 ) ) )
theorem Th57: :: JORDAN2C:57
for b
1 being
Natfor b
2 being
Realfor b
3 being
Subset of
(TOP-REAL b1)for b
4, b
5 being
Point of
(TOP-REAL b1) holds
not ( b
1 >= 2 & b
3 = (REAL b1) \ { b6 where B is Point of (TOP-REAL b1) : |.b6.| < b2 } & b
4 in b
3 & b
5 in b
3 & not for b
6 being
Real holds
( not b
4 = b
6 * b
5 & not b
5 = b
6 * b
4 ) & ( for b
6, b
7, b
8, b
9, b
10 being
Point of
(TOP-REAL b1) holds
not ( b
6 in b
3 & b
7 in b
3 & b
8 in b
3 & b
9 in b
3 & b
10 in b
3 &
LSeg b
4,b
6 c= b
3 &
LSeg b
6,b
7 c= b
3 &
LSeg b
7,b
8 c= b
3 &
LSeg b
8,b
9 c= b
3 &
LSeg b
9,b
10 c= b
3 &
LSeg b
10,b
5 c= b
3 ) ) )
theorem Th58: :: JORDAN2C:58
theorem Th59: :: JORDAN2C:59
theorem Th60: :: JORDAN2C:60
theorem Th61: :: JORDAN2C:61
theorem Th62: :: JORDAN2C:62
theorem Th63: :: JORDAN2C:63
theorem Th64: :: JORDAN2C:64
theorem Th65: :: JORDAN2C:65
theorem Th66: :: JORDAN2C:66
theorem Th67: :: JORDAN2C:67
theorem Th68: :: JORDAN2C:68
theorem Th69: :: JORDAN2C:69
theorem Th70: :: JORDAN2C:70
theorem Th71: :: JORDAN2C:71
theorem Th72: :: JORDAN2C:72
theorem Th73: :: JORDAN2C:73
theorem Th74: :: JORDAN2C:74
theorem Th75: :: JORDAN2C:75
theorem Th76: :: JORDAN2C:76
theorem Th77: :: JORDAN2C:77
theorem Th78: :: JORDAN2C:78
theorem Th79: :: JORDAN2C:79
theorem Th80: :: JORDAN2C:80
theorem Th81: :: JORDAN2C:81
theorem Th82: :: JORDAN2C:82
theorem Th83: :: JORDAN2C:83
theorem Th84: :: JORDAN2C:84
theorem Th85: :: JORDAN2C:85
theorem Th86: :: JORDAN2C:86
theorem Th87: :: JORDAN2C:87
theorem Th88: :: JORDAN2C:88
theorem Th89: :: JORDAN2C:89
theorem Th90: :: JORDAN2C:90
theorem Th91: :: JORDAN2C:91
theorem Th92: :: JORDAN2C:92
:: deftheorem Def9 JORDAN2C:def 9 :
canceled;
:: deftheorem Def10 defines pi JORDAN2C:def 10 :
theorem Th93: :: JORDAN2C:93
theorem Th94: :: JORDAN2C:94
theorem Th95: :: JORDAN2C:95
theorem Th96: :: JORDAN2C:96
theorem Th97: :: JORDAN2C:97
theorem Th98: :: JORDAN2C:98
theorem Th99: :: JORDAN2C:99
theorem Th100: :: JORDAN2C:100
theorem Th101: :: JORDAN2C:101
theorem Th102: :: JORDAN2C:102
theorem Th103: :: JORDAN2C:103
theorem Th104: :: JORDAN2C:104
theorem Th105: :: JORDAN2C:105
theorem Th106: :: JORDAN2C:106
theorem Th107: :: JORDAN2C:107
theorem Th108: :: JORDAN2C:108
theorem Th109: :: JORDAN2C:109
theorem Th110: :: JORDAN2C:110
theorem Th111: :: JORDAN2C:111
theorem Th112: :: JORDAN2C:112
theorem Th113: :: JORDAN2C:113
theorem Th114: :: JORDAN2C:114
theorem Th115: :: JORDAN2C:115
theorem Th116: :: JORDAN2C:116
theorem Th117: :: JORDAN2C:117
theorem Th118: :: JORDAN2C:118
theorem Th119: :: JORDAN2C:119
theorem Th120: :: JORDAN2C:120
theorem Th121: :: JORDAN2C:121
theorem Th122: :: JORDAN2C:122
theorem Th123: :: JORDAN2C:123
theorem Th124: :: JORDAN2C:124
theorem Th125: :: JORDAN2C:125
theorem Th126: :: JORDAN2C:126
theorem Th127: :: JORDAN2C:127
Lemma115:
for b1 being Point of (TOP-REAL 2) holds
b1 is Point of (Euclid 2)
theorem Th128: :: JORDAN2C:128
theorem Th129: :: JORDAN2C:129
theorem Th130: :: JORDAN2C:130
theorem Th131: :: JORDAN2C:131
theorem Th132: :: JORDAN2C:132
theorem Th133: :: JORDAN2C:133
theorem Th134: :: JORDAN2C:134
theorem Th135: :: JORDAN2C:135
theorem Th136: :: JORDAN2C:136
theorem Th137: :: JORDAN2C:137