:: JORDAN16 semantic presentation
theorem Th1: :: JORDAN16:1
theorem Th2: :: JORDAN16:2
for b
1, b
2, b
3, b
4 being
set holds
( b
1 in b
4 & b
2 in b
4 & b
3 in b
4 implies
{b1,b2,b3} c= b
4 )
theorem Th3: :: JORDAN16:3
theorem Th4: :: JORDAN16:4
theorem Th5: :: JORDAN16:5
theorem Th6: :: JORDAN16:6
theorem Th7: :: JORDAN16:7
theorem Th8: :: JORDAN16:8
theorem Th9: :: JORDAN16:9
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
( b
1 is_an_arc_of b
2,b
3 &
LE b
4,b
5,b
1,b
2,b
3 implies ( b
4 in Segment b
1,b
2,b
3,b
4,b
5 & b
5 in Segment b
1,b
2,b
3,b
4,b
5 ) )
theorem Th10: :: JORDAN16:10
theorem Th11: :: JORDAN16:11
theorem Th12: :: JORDAN16:12
theorem Th13: :: JORDAN16:13
theorem Th14: :: JORDAN16:14
theorem Th15: :: JORDAN16:15
theorem Th16: :: JORDAN16:16
theorem Th17: :: JORDAN16:17
theorem Th18: :: JORDAN16:18
theorem Th19: :: JORDAN16:19
theorem Th20: :: JORDAN16:20
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
not ( b
1 is_an_arc_of b
2,b
3 &
LE b
4,b
5,b
1,b
2,b
3 & ( for b
6 being
Function of
I[01] ,
((TOP-REAL 2) | b1)for b
7, b
8 being
Real holds
not ( b
6 is_homeomorphism & b
6 . 0
= b
2 & b
6 . 1
= b
3 & b
6 . b
7 = b
4 & b
6 . b
8 = b
5 & 0
<= b
7 & b
7 <= b
8 & b
8 <= 1 ) ) )
theorem Th21: :: JORDAN16:21
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
not ( b
1 is_an_arc_of b
2,b
3 &
LE b
4,b
5,b
1,b
2,b
3 & b
4 <> b
5 & ( for b
6 being
Function of
I[01] ,
((TOP-REAL 2) | b1)for b
7, b
8 being
Real holds
not ( b
6 is_homeomorphism & b
6 . 0
= b
2 & b
6 . 1
= b
3 & b
6 . b
7 = b
4 & b
6 . b
8 = b
5 & 0
<= b
7 & b
7 < b
8 & b
8 <= 1 ) ) )
theorem Th22: :: JORDAN16:22
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
not ( b
1 is_an_arc_of b
2,b
3 &
LE b
4,b
5,b
1,b
2,b
3 &
Segment b
1,b
2,b
3,b
4,b
5 is
empty )
theorem Th23: :: JORDAN16:23
:: deftheorem Def1 defines continuous JORDAN16:def 1 :
:: deftheorem Def2 defines continuous JORDAN16:def 2 :
:: deftheorem Def3 defines AffineMap JORDAN16:def 3 :
theorem Th24: :: JORDAN16:24
theorem Th25: :: JORDAN16:25
theorem Th26: :: JORDAN16:26
theorem Th27: :: JORDAN16:27
theorem Th28: :: JORDAN16:28
theorem Th29: :: JORDAN16:29
theorem Th30: :: JORDAN16:30
theorem Th31: :: JORDAN16:31
theorem Th32: :: JORDAN16:32
theorem Th33: :: JORDAN16:33
theorem Th34: :: JORDAN16:34
theorem Th35: :: JORDAN16:35
theorem Th36: :: JORDAN16:36
for b
1 being
Subset of
(TOP-REAL 2)for b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
( b
1 is_an_arc_of b
2,b
3 &
LE b
4,b
5,b
1,b
2,b
3 & b
4 <> b
5 implies
Segment b
1,b
2,b
3,b
4,b
5 is_an_arc_of b
4,b
5 )
theorem Th37: :: JORDAN16:37
theorem Th38: :: JORDAN16:38
theorem Th39: :: JORDAN16:39