:: JORDAN5B semantic presentation
theorem Th1: :: JORDAN5B:1
for b
1 being
Nat holds
not ( 1
<= b
1 & not b
1 -' 1
< b
1 )
theorem Th2: :: JORDAN5B:2
for b
1, b
2 being
Nat holds
( b
1 + 1
<= b
2 implies 1
<= b
2 -' b
1 )
theorem Th3: :: JORDAN5B:3
for b
1, b
2 being
Nat holds
( 1
<= b
1 & 1
<= b
2 implies
(b2 -' b1) + 1
<= b
2 )
Lemma2:
for b1 being real number holds
( 0 <= b1 & b1 <= 1 implies ( 0 <= 1 - b1 & 1 - b1 <= 1 ) )
theorem Th4: :: JORDAN5B:4
theorem Th5: :: JORDAN5B:5
theorem Th6: :: JORDAN5B:6
Lemma3:
for b1 being Point of I[01] holds
b1 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 b
1, b
2, b
3, b
4, b
5 being
Point of
(TOP-REAL 2) holds
( b
1 <> b
2 &
LE b
3,b
4,b
1,b
2 &
LE b
4,b
5,b
1,b
2 implies
LE b
3,b
5,b
1,b
2 )
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
Lemma10:
for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
not ( b2 in L~ b1 & b3 in L~ b1 & b2 <> b1 . (len b1) & b3 <> b1 . (len b1) & b1 is_S-Seq & not b2 in L~ (L_Cut b1,b3) & not b3 in L~ (L_Cut b1,b2) )
theorem Th25: :: JORDAN5B:25
theorem Th26: :: JORDAN5B:26
Lemma12:
for b1 being FinSequence of (TOP-REAL 2)
for b2, b3 being Point of (TOP-REAL 2) holds
( b2 in L~ b1 & b3 in L~ b1 & not ( not Index b2,b1 < Index b3,b1 & not ( Index b2,b1 = Index b3,b1 & LE b2,b3,b1 /. (Index b2,b1),b1 /. ((Index b2,b1) + 1) ) ) & b2 <> b3 implies L~ (B_Cut b1,b2,b3) c= L~ b1 )
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 b
1 being
FinSequence of
(TOP-REAL 2)for b
2, b
3 being
Point of
(TOP-REAL 2) holds
( b
2 in L~ b
1 & b
3 in L~ b
1 & not ( not
Index b
2,b
1 < Index b
3,b
1 & not (
Index b
2,b
1 = Index b
3,b
1 &
LE b
2,b
3,b
1 /. (Index b2,b1),b
1 /. ((Index b2,b1) + 1) ) ) & b
2 <> b
3 implies
L~ (B_Cut b1,b2,b3) c= L~ b
1 )
by Lemma12;