begin
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
Lm1:
the carrier of (TOP-REAL 2) = REAL 2
by EUCLID:25;
theorem
begin
theorem
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th13:
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
theorem
theorem
theorem
theorem
for
i,
j being
Element of
NAT for
C being
compact non
horizontal non
vertical Subset of 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:
for
i,
j,
n being
Element of
NAT for
C being
compact non
horizontal non
vertical Subset of 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:
for
i,
j,
n being
Element of
NAT for
C being
compact non
horizontal non
vertical Subset of 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
for
C being
compact non
horizontal non
vertical Subset of
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 )
begin
theorem Th22:
theorem
theorem Th24:
theorem Th25:
theorem Th26:
theorem Th27:
theorem
theorem Th29:
theorem Th30:
theorem Th31:
theorem
theorem Th33:
theorem Th34:
theorem Th35:
theorem Th36:
theorem Th37:
theorem Th38:
theorem Th39:
theorem Th40:
theorem Th41:
theorem Th42:
theorem Th43:
theorem Th44:
theorem
theorem Th46:
theorem Th47:
theorem
theorem Th49:
theorem
theorem
theorem
theorem
theorem