:: JORDAN5B semantic presentation
theorem Th1: :: JORDAN5B:1
theorem Th2: :: JORDAN5B:2
theorem Th3: :: JORDAN5B:3
Lemma27:
for r being real number st 0 <= r & r <= 1 holds
( 0 <= 1 - r & 1 - r <= 1 )
theorem Th4: :: JORDAN5B:4
theorem Th5: :: JORDAN5B:5
theorem Th6: :: JORDAN5B:6
Lemma39:
for P being Point of I[01] holds P is Real
theorem Th7: :: JORDAN5B:7
theorem Th8: :: JORDAN5B:8
theorem Th9: :: JORDAN5B:9
theorem Th10: :: JORDAN5B:10
theorem Th11: :: JORDAN5B:11
theorem Th12: :: JORDAN5B:12
for
p1,
p2,
q1,
q2,
q3 being
Point of
(TOP-REAL 2) st
p1 <> p2 &
LE q1,
q2,
p1,
p2 &
LE q2,
q3,
p1,
p2 holds
LE q1,
q3,
p1,
p2
theorem Th13: :: JORDAN5B:13
theorem Th14: :: JORDAN5B:14
theorem Th15: :: JORDAN5B:15
theorem Th16: :: JORDAN5B:16
theorem Th17: :: JORDAN5B:17
theorem Th18: :: JORDAN5B:18
canceled;
theorem Th19: :: JORDAN5B:19
canceled;
theorem Th20: :: JORDAN5B:20
canceled;
theorem Th21: :: JORDAN5B:21
theorem Th22: :: JORDAN5B:22
theorem Th23: :: JORDAN5B:23
theorem Th24: :: JORDAN5B:24
Lemma313:
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & p <> f . (len f) & q <> f . (len f) & f is_S-Seq & not p in L~ (L_Cut f,q) holds
q in L~ (L_Cut f,p)
theorem Th25: :: JORDAN5B:25
theorem Th26: :: JORDAN5B:26
Lemma315:
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & ( Index p,f < Index q,f or ( Index p,f = Index q,f & LE p,q,f /. (Index p,f),f /. ((Index p,f) + 1) ) ) & p <> q holds
L~ (B_Cut f,p,q) c= L~ f
theorem Th27: :: JORDAN5B:27
theorem Th28: :: JORDAN5B:28
theorem Th29: :: JORDAN5B:29
theorem Th30: :: JORDAN5B:30
theorem Th31: :: JORDAN5B:31
theorem Th32: :: JORDAN5B:32
theorem Th33: :: JORDAN5B:33
theorem Th34: :: JORDAN5B:34
theorem Th35: :: JORDAN5B:35
theorem Th36: :: JORDAN5B:36
for
f being
FinSequence of
(TOP-REAL 2) for
p,
q being
Point of
(TOP-REAL 2) st
p in L~ f &
q in L~ f & (
Index p,
f < Index q,
f or (
Index p,
f = Index q,
f &
LE p,
q,
f /. (Index p,f),
f /. ((Index p,f) + 1) ) ) &
p <> q holds
L~ (B_Cut f,p,q) c= L~ f by ;