begin
theorem Th1:
for
k,
n,
i,
j being
Element of
NAT for
C being
connected compact non
horizontal non
vertical Subset of st 1
<= k &
k + 1
<= len (Cage C,n) &
[i,j] in Indices (Gauge C,n) &
[i,(j + 1)] in Indices (Gauge C,n) &
(Cage C,n) /. k = (Gauge C,n) * i,
j &
(Cage C,n) /. (k + 1) = (Gauge C,n) * i,
(j + 1) holds
i < len (Gauge C,n)
theorem Th2:
for
k,
n,
i,
j being
Element of
NAT for
C being
connected compact non
horizontal non
vertical Subset of st 1
<= k &
k + 1
<= len (Cage C,n) &
[i,j] in Indices (Gauge C,n) &
[i,(j + 1)] in Indices (Gauge C,n) &
(Cage C,n) /. k = (Gauge C,n) * i,
(j + 1) &
(Cage C,n) /. (k + 1) = (Gauge C,n) * i,
j holds
i > 1
theorem Th3:
for
k,
n,
i,
j being
Element of
NAT for
C being
connected compact non
horizontal non
vertical Subset of st 1
<= k &
k + 1
<= len (Cage C,n) &
[i,j] in Indices (Gauge C,n) &
[(i + 1),j] in Indices (Gauge C,n) &
(Cage C,n) /. k = (Gauge C,n) * i,
j &
(Cage C,n) /. (k + 1) = (Gauge C,n) * (i + 1),
j holds
j > 1
theorem Th4:
for
k,
n,
i,
j being
Element of
NAT for
C being
connected compact non
horizontal non
vertical Subset of st 1
<= k &
k + 1
<= len (Cage C,n) &
[i,j] in Indices (Gauge C,n) &
[(i + 1),j] in Indices (Gauge C,n) &
(Cage C,n) /. k = (Gauge C,n) * (i + 1),
j &
(Cage C,n) /. (k + 1) = (Gauge C,n) * i,
j holds
j < width (Gauge C,n)
theorem Th5:
theorem Th6:
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
theorem Th12:
theorem Th13:
:: deftheorem defines UBD-Family JORDAN10:def 1 :
:: deftheorem defines BDD-Family JORDAN10:def 2 :
theorem Th14:
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem