:: TOPREAL3 semantic presentation
theorem Th1: :: TOPREAL3:1
canceled;
theorem Th2: :: TOPREAL3:2
canceled;
theorem Th3: :: TOPREAL3:3
for b
1, b
2 being
real number holds
( b
1 < b
2 implies ( b
1 < (b1 + b2) / 2 &
(b1 + b2) / 2
< b
2 ) )
Lemma2:
for b1 being Nat holds the carrier of (Euclid b1) = REAL b1
theorem Th4: :: TOPREAL3:4
canceled;
theorem Th5: :: TOPREAL3:5
canceled;
theorem Th6: :: TOPREAL3:6
theorem Th7: :: TOPREAL3:7
theorem Th8: :: TOPREAL3:8
theorem Th9: :: TOPREAL3:9
theorem Th10: :: TOPREAL3:10
theorem Th11: :: TOPREAL3:11
theorem Th12: :: TOPREAL3:12
theorem Th13: :: TOPREAL3:13
theorem Th14: :: TOPREAL3:14
canceled;
theorem Th15: :: TOPREAL3:15
theorem Th16: :: TOPREAL3:16
theorem Th17: :: TOPREAL3:17
theorem Th18: :: TOPREAL3:18
theorem Th19: :: TOPREAL3:19
theorem Th20: :: TOPREAL3:20
theorem Th21: :: TOPREAL3:21
theorem Th22: :: TOPREAL3:22
canceled;
theorem Th23: :: TOPREAL3:23
theorem Th24: :: TOPREAL3:24
theorem Th25: :: TOPREAL3:25
theorem Th26: :: TOPREAL3:26
theorem Th27: :: TOPREAL3:27
theorem Th28: :: TOPREAL3:28
theorem Th29: :: TOPREAL3:29
for b
1, b
2, b
3 being
Point of
(TOP-REAL 2)for b
4, b
5, b
6, b
7, b
8 being
real number for b
9 being
Point of
(Euclid 2) holds
( b
9 = b
1 & b
1 = |[b4,b5]| & b
2 = |[b6,b7]| & b
3 = |[b6,b5]| & b
2 in Ball b
9,b
8 implies b
3 in Ball b
9,b
8 )
theorem Th30: :: TOPREAL3:30
theorem Th31: :: TOPREAL3:31
theorem Th32: :: TOPREAL3:32
for b
1, b
2, b
3, b
4, b
5 being
real number for b
6 being
Point of
(Euclid 2) holds
not ( b
1 <> b
2 & b
3 <> b
4 &
|[b1,b4]| in Ball b
6,b
5 &
|[b2,b3]| in Ball b
6,b
5 & not
|[b1,b3]| in Ball b
6,b
5 & not
|[b2,b4]| in Ball b
6,b
5 )
theorem Th33: :: TOPREAL3:33
theorem Th34: :: TOPREAL3:34
theorem Th35: :: TOPREAL3:35
theorem Th36: :: TOPREAL3:36
theorem Th37: :: TOPREAL3:37
theorem Th38: :: TOPREAL3:38
theorem Th39: :: TOPREAL3:39
theorem Th40: :: TOPREAL3:40
theorem Th41: :: TOPREAL3:41
theorem Th42: :: TOPREAL3:42
theorem Th43: :: TOPREAL3:43
theorem Th44: :: TOPREAL3:44
theorem Th45: :: TOPREAL3:45
theorem Th46: :: TOPREAL3:46
theorem Th47: :: TOPREAL3:47
theorem Th48: :: TOPREAL3:48
theorem Th49: :: TOPREAL3:49
theorem Th50: :: TOPREAL3:50
for b
1, b
2, b
3 being
Point of
(TOP-REAL 2)for b
4 being
real number for b
5 being
Point of
(Euclid 2) holds
( not b
1 in Ball b
5,b
4 & b
2 in Ball b
5,b
4 &
|[(b2 `1 ),(b3 `2 )]| in Ball b
5,b
4 & b
3 in Ball b
5,b
4 & not
|[(b2 `1 ),(b3 `2 )]| in LSeg b
1,b
2 & b
1 `1 = b
2 `1 & b
2 `1 <> b
3 `1 & b
2 `2 <> b
3 `2 implies
((LSeg b2,|[(b2 `1 ),(b3 `2 )]|) \/ (LSeg |[(b2 `1 ),(b3 `2 )]|,b3)) /\ (LSeg b1,b2) = {b2} )
theorem Th51: :: TOPREAL3:51
for b
1, b
2, b
3 being
Point of
(TOP-REAL 2)for b
4 being
real number for b
5 being
Point of
(Euclid 2) holds
( not b
1 in Ball b
5,b
4 & b
2 in Ball b
5,b
4 &
|[(b3 `1 ),(b2 `2 )]| in Ball b
5,b
4 & b
3 in Ball b
5,b
4 & not
|[(b3 `1 ),(b2 `2 )]| in LSeg b
1,b
2 & b
1 `2 = b
2 `2 & b
2 `1 <> b
3 `1 & b
2 `2 <> b
3 `2 implies
((LSeg b2,|[(b3 `1 ),(b2 `2 )]|) \/ (LSeg |[(b3 `1 ),(b2 `2 )]|,b3)) /\ (LSeg b1,b2) = {b2} )