:: JORDAN5A semantic presentation
theorem Th1: :: JORDAN5A:1
Lemma2:
for b1 being Nat holds
( the carrier of (Euclid b1) = REAL b1 & the carrier of (TOP-REAL b1) = REAL b1 )
theorem Th2: :: JORDAN5A:2
theorem Th3: :: JORDAN5A:3
theorem Th4: :: JORDAN5A:4
Lemma6:
for b1 being Nat holds TOP-REAL b1 is arcwise_connected
theorem Th5: :: JORDAN5A:5
Lemma7:
for b1 being Subset of REAL holds
( b1 is open implies b1 in Family_open_set RealSpace )
Lemma8:
for b1 being Subset of REAL holds
( b1 in Family_open_set RealSpace implies b1 is open )
theorem Th6: :: JORDAN5A:6
theorem Th7: :: JORDAN5A:7
theorem Th8: :: JORDAN5A:8
Lemma11:
for b1 being one-to-one continuous Function of R^1 ,R^1
for b2 being PartFunc of REAL , REAL holds
not ( b1 = b2 & not b2 is_increasing_on [.0,1.] & not b2 is_decreasing_on [.0,1.] )
theorem Th9: :: JORDAN5A:9
theorem Th10: :: JORDAN5A:10
theorem Th11: :: JORDAN5A:11
theorem Th12: :: JORDAN5A:12
theorem Th13: :: JORDAN5A:13
Lemma16:
for b1, b2, b3 being real number holds
( b1 <= b2 implies ( b3 in the carrier of (Closed-Interval-TSpace b1,b2) iff ( b1 <= b3 & b3 <= b2 ) ) )
Lemma17:
for b1, b2, b3, b4 being Real
for b5 being Function of (Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace b3,b4)
for b6 being Point of (Closed-Interval-TSpace b1,b2)
for b7 being PartFunc of REAL , REAL
for b8 being Real holds
( b1 < b2 & b3 < b4 & b5 is_continuous_at b6 & b6 <> b1 & b6 <> b2 & b5 . b1 = b3 & b5 . b2 = b4 & b5 is one-to-one & b5 = b7 & b6 = b8 implies b7 is_continuous_in b8 )
theorem Th14: :: JORDAN5A:14
for b
1, b
2, b
3, b
4, b
5 being
Realfor b
6 being
Function of
(Closed-Interval-TSpace b1,b2),
(Closed-Interval-TSpace b3,b4)for b
7 being
Point of
(Closed-Interval-TSpace b1,b2)for b
8 being
PartFunc of
REAL ,
REAL holds
( b
1 < b
2 & b
3 < b
4 & b
6 is_continuous_at b
7 & b
6 . b
1 = b
3 & b
6 . b
2 = b
4 & b
6 is
one-to-one & b
6 = b
8 & b
7 = b
5 implies b
8 | [.b1,b2.] is_continuous_in b
5 )
theorem Th15: :: JORDAN5A:15
theorem Th16: :: JORDAN5A:16
for b
1, b
2, b
3, b
4 being
Realfor b
5 being
Function of
(Closed-Interval-TSpace b1,b2),
(Closed-Interval-TSpace b3,b4) holds
( b
1 < b
2 & b
3 < b
4 & b
5 is
continuous & b
5 is
one-to-one & b
5 . b
1 = b
3 & b
5 . b
2 = b
4 implies for b
6, b
7 being
Point of
(Closed-Interval-TSpace b1,b2)for b
8, b
9, b
10, b
11 being
Real holds
not ( b
6 = b
8 & b
7 = b
9 & b
8 < b
9 & b
10 = b
5 . b
6 & b
11 = b
5 . b
7 & not b
10 < b
11 ) )
theorem Th17: :: JORDAN5A:17
theorem Th18: :: JORDAN5A:18
theorem Th19: :: JORDAN5A:19
theorem Th20: :: JORDAN5A:20
theorem Th21: :: JORDAN5A:21
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
Realfor b
9 being
Function of
(Closed-Interval-TSpace b1,b2),
(Closed-Interval-TSpace b3,b4) holds
( b
1 < b
2 & b
3 < b
4 & b
5 < b
6 & b
1 <= b
5 & b
6 <= b
2 & b
9 is_homeomorphism & b
9 . b
1 = b
3 & b
9 . b
2 = b
4 & b
7 = b
9 . b
5 & b
8 = b
9 . b
6 implies b
9 .: [.b5,b6.] = [.b7,b8.] )
theorem Th22: :: JORDAN5A:22
theorem Th23: :: JORDAN5A:23