:: TOPREAL3 semantic presentation
theorem Th1: :: TOPREAL3:1
canceled;
theorem Th2: :: TOPREAL3:2
canceled;
theorem Th3: :: TOPREAL3:3
Lemma51:
for n being Element of NAT holds the carrier of (Euclid n) = REAL n
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
p1,
p2,
p being
Point of
(TOP-REAL 2) for
r1,
s1,
r2,
s2,
r being
real number for
u being
Point of
(Euclid 2) st
u = p1 &
p1 = |[r1,s1]| &
p2 = |[r2,s2]| &
p = |[r2,s1]| &
p2 in Ball u,
r holds
p in Ball u,
r
theorem Th30: :: TOPREAL3:30
theorem Th31: :: TOPREAL3:31
theorem Th32: :: TOPREAL3:32
for
r1,
s1,
s2,
r2,
r being
real number for
u being
Point of
(Euclid 2) st
r1 <> s1 &
s2 <> r2 &
|[r1,r2]| in Ball u,
r &
|[s1,s2]| in Ball u,
r & not
|[r1,s2]| in Ball u,
r holds
|[s1,r2]| in Ball u,
r
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
p1,
p,
q being
Point of
(TOP-REAL 2) for
r being
real number for
u being
Point of
(Euclid 2) st not
p1 in Ball u,
r &
p in Ball u,
r &
|[(p `1 ),(q `2 )]| in Ball u,
r &
q in Ball u,
r & not
|[(p `1 ),(q `2 )]| in LSeg p1,
p &
p1 `1 = p `1 &
p `1 <> q `1 &
p `2 <> q `2 holds
((LSeg p,|[(p `1 ),(q `2 )]|) \/ (LSeg |[(p `1 ),(q `2 )]|,q)) /\ (LSeg p1,p) = {p}
theorem Th51: :: TOPREAL3:51
for
p1,
p,
q being
Point of
(TOP-REAL 2) for
r being
real number for
u being
Point of
(Euclid 2) st not
p1 in Ball u,
r &
p in Ball u,
r &
|[(q `1 ),(p `2 )]| in Ball u,
r &
q in Ball u,
r & not
|[(q `1 ),(p `2 )]| in LSeg p1,
p &
p1 `2 = p `2 &
p `1 <> q `1 &
p `2 <> q `2 holds
((LSeg p,|[(q `1 ),(p `2 )]|) \/ (LSeg |[(q `1 ),(p `2 )]|,q)) /\ (LSeg p1,p) = {p}