Article JORDAN1D, MML version 4.99.1005
:: JORDAN1D:th 1
theorem
for b1, b2 being set
st for b3 being set
st b3 in b1
holds ex b4 being set st
b4 c= b2 & b3 c= union b4
holds union b1 c= union b2;
:: JORDAN1D:funcreg 1
registration
let a1 be non empty Element of NAT;
cluster 2 |^ a1 -> even;
end;
:: JORDAN1D:funcreg 2
registration
let a1 be even Element of NAT;
let a2 be non empty Element of NAT;
cluster a1 |^ a2 -> even;
end;
:: JORDAN1D:th 2
theorem
for b1 being Element of NAT
for b2 being real set
st b2 <> 0
holds (1 / b2) * (b2 |^ (b1 + 1)) = b2 |^ b1;
:: JORDAN1D:th 7
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non empty Element of bool the carrier of TOP-REAL 2
st 2 <= b1 & b1 < len Gauge(b5,b2) & 1 <= b3 & b3 <= len Gauge(b5,b2) & 1 <= b4 & b4 <= len Gauge(b5,b2 + 1)
holds ((Gauge(b5,b2)) *(b1,b3)) `1 = ((Gauge(b5,b2 + 1)) *((2 * b1) -' 2,b4)) `1;
:: JORDAN1D:th 8
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non empty Element of bool the carrier of TOP-REAL 2
st 2 <= b1 & b1 < len Gauge(b5,b2) & 1 <= b3 & b3 <= len Gauge(b5,b2) & 1 <= b4 & b4 <= len Gauge(b5,b2 + 1)
holds ((Gauge(b5,b2)) *(b3,b1)) `2 = ((Gauge(b5,b2 + 1)) *(b4,(2 * b1) -' 2)) `2;
:: JORDAN1D:th 9
theorem
for b1, b2, b3 being Element of NAT
for b4 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st 2 <= b1 & b1 + 1 < len Gauge(b4,b2) & 2 <= b3 & b3 + 1 < len Gauge(b4,b2)
holds cell(Gauge(b4,b2),b1,b3) = (((cell(Gauge(b4,b2 + 1),(2 * b1) -' 2,(2 * b3) -' 2)) \/ cell(Gauge(b4,b2 + 1),(2 * b1) -' 1,(2 * b3) -' 2)) \/ cell(Gauge(b4,b2 + 1),(2 * b1) -' 2,(2 * b3) -' 1)) \/ cell(Gauge(b4,b2 + 1),(2 * b1) -' 1,(2 * b3) -' 1);
:: JORDAN1D:th 10
theorem
for b1, b2, b3 being Element of NAT
for b4 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b5 being Element of NAT
st 2 <= b1 & b1 + 1 < len Gauge(b4,b2) & 2 <= b3 & b3 + 1 < len Gauge(b4,b2)
holds cell(Gauge(b4,b2),b1,b3) = union {cell(Gauge(b4,b2 + b5),b6,b7) where b6 is Element of NAT, b7 is Element of NAT: (((2 |^ b5) * b1) - (2 |^ (b5 + 1))) + 2 <= b6 &
b6 <= (((2 |^ b5) * b1) - (2 |^ b5)) + 1 &
(((2 |^ b5) * b3) - (2 |^ (b5 + 1))) + 2 <= b7 &
b7 <= (((2 |^ b5) * b3) - (2 |^ b5)) + 1};
:: JORDAN1D:th 11
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 < len Cage(b2,b1) &
N-max b2 in right_cell(Cage(b2,b1),b3,Gauge(b2,b1));
:: JORDAN1D:th 12
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 & b3 < len Cage(b2,b1) & N-max b2 in right_cell(Cage(b2,b1),b3);
:: JORDAN1D:th 13
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 < len Cage(b2,b1) &
E-min b2 in right_cell(Cage(b2,b1),b3,Gauge(b2,b1));
:: JORDAN1D:th 14
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 & b3 < len Cage(b2,b1) & E-min b2 in right_cell(Cage(b2,b1),b3);
:: JORDAN1D:th 15
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 < len Cage(b2,b1) &
E-max b2 in right_cell(Cage(b2,b1),b3,Gauge(b2,b1));
:: JORDAN1D:th 16
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 & b3 < len Cage(b2,b1) & E-max b2 in right_cell(Cage(b2,b1),b3);
:: JORDAN1D:th 17
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 < len Cage(b2,b1) &
S-min b2 in right_cell(Cage(b2,b1),b3,Gauge(b2,b1));
:: JORDAN1D:th 18
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 & b3 < len Cage(b2,b1) & S-min b2 in right_cell(Cage(b2,b1),b3);
:: JORDAN1D:th 19
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 < len Cage(b2,b1) &
S-max b2 in right_cell(Cage(b2,b1),b3,Gauge(b2,b1));
:: JORDAN1D:th 20
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 & b3 < len Cage(b2,b1) & S-max b2 in right_cell(Cage(b2,b1),b3);
:: JORDAN1D:th 21
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 < len Cage(b2,b1) &
W-min b2 in right_cell(Cage(b2,b1),b3,Gauge(b2,b1));
:: JORDAN1D:th 22
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 & b3 < len Cage(b2,b1) & W-min b2 in right_cell(Cage(b2,b1),b3);
:: JORDAN1D:th 23
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 < len Cage(b2,b1) &
W-max b2 in right_cell(Cage(b2,b1),b3,Gauge(b2,b1));
:: JORDAN1D:th 24
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 & b3 < len Cage(b2,b1) & W-max b2 in right_cell(Cage(b2,b1),b3);
:: JORDAN1D:th 25
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= len Gauge(b2,b1) &
N-min L~ Cage(b2,b1) = (Gauge(b2,b1)) *(b3,width Gauge(b2,b1));
:: JORDAN1D:th 26
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= len Gauge(b2,b1) &
N-max L~ Cage(b2,b1) = (Gauge(b2,b1)) *(b3,width Gauge(b2,b1));
:: JORDAN1D:th 27
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= len Gauge(b2,b1) &
(Gauge(b2,b1)) *(b3,width Gauge(b2,b1)) in proj2 Cage(b2,b1);
:: JORDAN1D:th 28
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= width Gauge(b2,b1) &
E-min L~ Cage(b2,b1) = (Gauge(b2,b1)) *(len Gauge(b2,b1),b3);
:: JORDAN1D:th 29
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= width Gauge(b2,b1) &
E-max L~ Cage(b2,b1) = (Gauge(b2,b1)) *(len Gauge(b2,b1),b3);
:: JORDAN1D:th 30
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= width Gauge(b2,b1) &
(Gauge(b2,b1)) *(len Gauge(b2,b1),b3) in proj2 Cage(b2,b1);
:: JORDAN1D:th 31
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= len Gauge(b2,b1) &
S-min L~ Cage(b2,b1) = (Gauge(b2,b1)) *(b3,1);
:: JORDAN1D:th 32
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= len Gauge(b2,b1) &
S-max L~ Cage(b2,b1) = (Gauge(b2,b1)) *(b3,1);
:: JORDAN1D:th 33
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= len Gauge(b2,b1) &
(Gauge(b2,b1)) *(b3,1) in proj2 Cage(b2,b1);
:: JORDAN1D:th 34
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= width Gauge(b2,b1) &
W-min L~ Cage(b2,b1) = (Gauge(b2,b1)) *(1,b3);
:: JORDAN1D:th 35
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= width Gauge(b2,b1) &
W-max L~ Cage(b2,b1) = (Gauge(b2,b1)) *(1,b3);
:: JORDAN1D:th 36
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
ex b3 being Element of NAT st
1 <= b3 &
b3 <= width Gauge(b2,b1) &
(Gauge(b2,b1)) *(1,b3) in proj2 Cage(b2,b1);