:: GOBRD14 semantic presentation
theorem Th1: :: GOBRD14:1
theorem Th2: :: GOBRD14:2
theorem Th3: :: GOBRD14:3
theorem Th4: :: GOBRD14:4
theorem Th5: :: GOBRD14:5
theorem Th6: :: GOBRD14:6
theorem Th7: :: GOBRD14:7
theorem Th8: :: GOBRD14:8
theorem Th9: :: GOBRD14:9
theorem Th10: :: GOBRD14:10
theorem Th11: :: GOBRD14:11
theorem Th12: :: GOBRD14:12
theorem Th13: :: GOBRD14:13
for
b1,
b2 being
Natfor
b3 being
Go-board st 1
<= b1 &
b1 < len b3 & 1
<= b2 &
b2 < width b3 holds
cell b3,
b1,
b2 = product (1,2 --> [.((b3 * b1,1) `1 ),((b3 * (b1 + 1),1) `1 ).],[.((b3 * 1,b2) `2 ),((b3 * 1,(b2 + 1)) `2 ).])
theorem Th14: :: GOBRD14:14
theorem Th15: :: GOBRD14:15
theorem Th16: :: GOBRD14:16
theorem Th17: :: GOBRD14:17
theorem Th18: :: GOBRD14:18
for
b1,
b2 being
Natfor
b3 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
b1 <= b2 holds
for
b4,
b5 being
Nat st 2
<= b4 &
b4 <= (len (Gauge b3,b1)) - 1 & 2
<= b5 &
b5 <= (len (Gauge b3,b1)) - 1 holds
ex
b6,
b7 being
Nat st
( 2
<= b6 &
b6 <= (len (Gauge b3,b2)) - 1 & 2
<= b7 &
b7 <= (len (Gauge b3,b2)) - 1 &
[b6,b7] in Indices (Gauge b3,b2) &
(Gauge b3,b1) * b4,
b5 = (Gauge b3,b2) * b6,
b7 &
b6 = 2
+ ((2 |^ (b2 -' b1)) * (b4 -' 2)) &
b7 = 2
+ ((2 |^ (b2 -' b1)) * (b5 -' 2)) )
theorem Th19: :: GOBRD14:19
for
b1,
b2,
b3 being
Natfor
b4 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
[b1,b2] in Indices (Gauge b4,b3) &
[b1,(b2 + 1)] in Indices (Gauge b4,b3) holds
dist ((Gauge b4,b3) * b1,b2),
((Gauge b4,b3) * b1,(b2 + 1)) = ((N-bound b4) - (S-bound b4)) / (2 |^ b3)
theorem Th20: :: GOBRD14:20
for
b1,
b2,
b3 being
Natfor
b4 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
[b1,b2] in Indices (Gauge b4,b3) &
[(b1 + 1),b2] in Indices (Gauge b4,b3) holds
dist ((Gauge b4,b3) * b1,b2),
((Gauge b4,b3) * (b1 + 1),b2) = ((E-bound b4) - (W-bound b4)) / (2 |^ b3)
theorem Th21: :: GOBRD14:21
for
b1 being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2)for
b2,
b3 being
real number st
b2 > 0 &
b3 > 0 holds
ex
b4 being
Nat st
( 1
< b4 &
dist ((Gauge b1,b4) * 1,1),
((Gauge b1,b4) * 1,2) < b2 &
dist ((Gauge b1,b4) * 1,1),
((Gauge b1,b4) * 2,1) < b3 )
theorem Th22: :: GOBRD14:22
theorem Th23: :: GOBRD14:23
theorem Th24: :: GOBRD14:24
theorem Th25: :: GOBRD14:25
theorem Th26: :: GOBRD14:26
theorem Th27: :: GOBRD14:27
theorem Th28: :: GOBRD14:28
theorem Th29: :: GOBRD14:29
theorem Th30: :: GOBRD14:30
theorem Th31: :: GOBRD14:31
theorem Th32: :: 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 Th45: :: GOBRD14:45
theorem Th46: :: GOBRD14:46
theorem Th47: :: GOBRD14:47
theorem Th48: :: GOBRD14:48
theorem Th49: :: GOBRD14:49
theorem Th50: :: GOBRD14:50
theorem Th51: :: GOBRD14:51
theorem Th52: :: GOBRD14:52
theorem Th53: :: GOBRD14:53
theorem Th54: :: GOBRD14:54