:: JORDAN21 semantic presentation
Lemma1:
dom proj2 = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;
Lemma2:
for b1 being real number
for b2 being Subset of (TOP-REAL 2) st b1 in proj2 .: b2 holds
ex b3 being Point of (TOP-REAL 2) st
( b3 in b2 & proj2 . b3 = b1 )
Lemma4:
for b1, b2, b3, b4 being set st b1 misses b4 & b2 misses b4 & b3 misses b4 holds
(b1 \/ b2) \/ b3 misses b4
theorem Th1: :: JORDAN21:1
theorem Th2: :: JORDAN21:2
theorem Th3: :: JORDAN21:3
theorem Th4: :: JORDAN21:4
theorem Th5: :: JORDAN21:5
theorem Th6: :: JORDAN21:6
theorem Th7: :: JORDAN21:7
theorem Th8: :: JORDAN21:8
theorem Th9: :: JORDAN21:9
theorem Th10: :: JORDAN21:10
theorem Th11: :: JORDAN21:11
for
b1 being
Subset of the
carrier of
(TOP-REAL 2)for
b2,
b3,
b4,
b5 being
Point of
(TOP-REAL 2) st
b1 is_an_arc_of b2,
b3 &
b2 <> b4 &
b3 <> b5 holds
( not
b2 in Segment b1,
b2,
b3,
b4,
b5 & not
b3 in Segment b1,
b2,
b3,
b4,
b5 )
:: deftheorem Def1 defines with_the_max_arc JORDAN21:def 1 :
Lemma11:
for b1 being Simple_closed_curve holds Upper_Middle_Point b1 in b1
by JORDAN6:83;
theorem Th12: :: JORDAN21:12
theorem Th13: :: JORDAN21:13
theorem Th14: :: JORDAN21:14
canceled;
theorem Th15: :: JORDAN21:15
canceled;
theorem Th16: :: JORDAN21:16
canceled;
theorem Th17: :: JORDAN21:17
canceled;
theorem Th18: :: JORDAN21:18
canceled;
theorem Th19: :: JORDAN21:19
canceled;
theorem Th20: :: JORDAN21:20
canceled;
theorem Th21: :: JORDAN21:21
canceled;
theorem Th22: :: JORDAN21:22
canceled;
theorem Th23: :: JORDAN21:23
theorem Th24: :: JORDAN21:24
theorem Th25: :: JORDAN21:25
theorem Th26: :: JORDAN21:26
theorem Th27: :: JORDAN21:27
theorem Th28: :: JORDAN21:28
theorem Th29: :: JORDAN21:29
theorem Th30: :: JORDAN21:30
theorem Th31: :: JORDAN21:31
theorem Th32: :: JORDAN21:32
:: deftheorem Def2 defines UMP JORDAN21:def 2 :
:: deftheorem Def3 defines LMP JORDAN21:def 3 :
theorem Th33: :: JORDAN21:33
theorem Th34: :: JORDAN21:34
theorem Th35: :: JORDAN21:35
theorem Th36: :: JORDAN21:36
theorem Th37: :: JORDAN21:37
theorem Th38: :: JORDAN21:38
theorem Th39: :: JORDAN21:39
theorem Th40: :: JORDAN21:40
theorem Th41: :: JORDAN21:41
theorem Th42: :: JORDAN21:42
theorem Th43: :: JORDAN21:43
theorem Th44: :: JORDAN21:44
theorem Th45: :: JORDAN21:45
theorem Th46: :: JORDAN21:46
theorem Th47: :: JORDAN21:47
theorem Th48: :: JORDAN21:48
theorem Th49: :: JORDAN21:49
theorem Th50: :: JORDAN21:50
theorem Th51: :: JORDAN21:51
theorem Th52: :: JORDAN21:52
theorem Th53: :: JORDAN21:53
theorem Th54: :: JORDAN21:54
theorem Th55: :: JORDAN21:55
theorem Th56: :: JORDAN21:56
theorem Th57: :: JORDAN21:57
theorem Th58: :: JORDAN21:58
theorem Th59: :: JORDAN21:59
theorem Th60: :: JORDAN21:60
theorem Th61: :: JORDAN21:61
theorem Th62: :: JORDAN21:62
theorem Th63: :: JORDAN21:63