:: Properties of Left-, and Right Components
:: by Artur Korni{\l}owicz
::
:: Received May 5, 1999
:: Copyright (c) 1999 Association of Mizar Users
theorem :: GOBRD14:1
canceled;
theorem :: GOBRD14:2
canceled;
theorem :: GOBRD14:3
canceled;
theorem :: GOBRD14:4
canceled;
theorem :: GOBRD14:5
canceled;
theorem :: GOBRD14:6
canceled;
theorem :: GOBRD14:7
theorem :: GOBRD14:8
theorem :: GOBRD14:9
canceled;
theorem :: GOBRD14:10
canceled;
theorem :: GOBRD14:11
canceled;
theorem :: GOBRD14:12
canceled;
theorem Th13: :: GOBRD14:13
for
i,
j being
Element of
NAT for
G being
Go-board st 1
<= i &
i < len G & 1
<= j &
j < width G holds
cell G,
i,
j = product (1,2 --> [.((G * i,1) `1 ),((G * (i + 1),1) `1 ).],[.((G * 1,j) `2 ),((G * 1,(j + 1)) `2 ).])
theorem :: GOBRD14:14
theorem :: GOBRD14:15
theorem :: GOBRD14:16
theorem :: GOBRD14:17
theorem :: GOBRD14:18
for
i,
j being
Element of
NAT for
C being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
i <= j holds
for
a,
b being
Element of
NAT st 2
<= a &
a <= (len (Gauge C,i)) - 1 & 2
<= b &
b <= (len (Gauge C,i)) - 1 holds
ex
c,
d being
Element of
NAT st
( 2
<= c &
c <= (len (Gauge C,j)) - 1 & 2
<= d &
d <= (len (Gauge C,j)) - 1 &
[c,d] in Indices (Gauge C,j) &
(Gauge C,i) * a,
b = (Gauge C,j) * c,
d &
c = 2
+ ((2 |^ (j -' i)) * (a -' 2)) &
d = 2
+ ((2 |^ (j -' i)) * (b -' 2)) )
theorem Th19: :: GOBRD14:19
for
i,
j,
n being
Element of
NAT for
C being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
[i,j] in Indices (Gauge C,n) &
[i,(j + 1)] in Indices (Gauge C,n) holds
dist ((Gauge C,n) * i,j),
((Gauge C,n) * i,(j + 1)) = ((N-bound C) - (S-bound C)) / (2 |^ n)
theorem Th20: :: GOBRD14:20
for
i,
j,
n being
Element of
NAT for
C being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
[i,j] in Indices (Gauge C,n) &
[(i + 1),j] in Indices (Gauge C,n) holds
dist ((Gauge C,n) * i,j),
((Gauge C,n) * (i + 1),j) = ((E-bound C) - (W-bound C)) / (2 |^ n)
theorem :: GOBRD14:21
for
C being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) for
r,
t being
real number st
r > 0 &
t > 0 holds
ex
n being
Element of
NAT st
( 1
< n &
dist ((Gauge C,n) * 1,1),
((Gauge C,n) * 1,2) < r &
dist ((Gauge C,n) * 1,1),
((Gauge C,n) * 2,1) < t )
theorem Th22: :: GOBRD14:22
theorem :: GOBRD14:23
theorem Th24: :: GOBRD14:24
theorem Th25: :: GOBRD14:25
theorem Th26: :: GOBRD14:26
theorem Th27: :: GOBRD14:27
theorem :: GOBRD14:28
theorem Th29: :: GOBRD14:29
theorem Th30: :: GOBRD14:30
theorem Th31: :: GOBRD14:31
theorem :: GOBRD14:32
theorem Th33: :: GOBRD14:33
theorem Th34: :: GOBRD14:34
theorem Th35: :: GOBRD14:35
theorem Th36: :: GOBRD14:36
theorem Th37: :: GOBRD14:37
theorem Th38: :: GOBRD14:38
theorem Th39: :: GOBRD14:39
theorem Th40: :: GOBRD14:40
theorem Th41: :: GOBRD14:41
theorem Th42: :: GOBRD14:42
theorem Th43: :: GOBRD14:43
theorem Th44: :: GOBRD14:44
theorem :: GOBRD14:45
theorem Th46: :: GOBRD14:46
theorem Th47: :: GOBRD14:47
theorem :: GOBRD14:48
theorem Th49: :: GOBRD14:49
theorem :: GOBRD14:50
theorem :: GOBRD14:51
theorem :: GOBRD14:52
theorem :: GOBRD14:53
theorem :: GOBRD14:54