Article JORDAN10, MML version 4.99.1005
:: JORDAN10:exreg 1
registration
cluster connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
end;
:: JORDAN10:th 1
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st 1 <= b1 &
b1 + 1 <= len Cage(b5,b2) &
[b3,b4] in Indices Gauge(b5,b2) &
[b3,b4 + 1] in Indices Gauge(b5,b2) &
(Cage(b5,b2)) /. b1 = (Gauge(b5,b2)) *(b3,b4) &
(Cage(b5,b2)) /. (b1 + 1) = (Gauge(b5,b2)) *(b3,b4 + 1)
holds b3 < len Gauge(b5,b2);
:: JORDAN10:th 2
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st 1 <= b1 &
b1 + 1 <= len Cage(b5,b2) &
[b3,b4] in Indices Gauge(b5,b2) &
[b3,b4 + 1] in Indices Gauge(b5,b2) &
(Cage(b5,b2)) /. b1 = (Gauge(b5,b2)) *(b3,b4 + 1) &
(Cage(b5,b2)) /. (b1 + 1) = (Gauge(b5,b2)) *(b3,b4)
holds 1 < b3;
:: JORDAN10:th 3
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st 1 <= b1 &
b1 + 1 <= len Cage(b5,b2) &
[b3,b4] in Indices Gauge(b5,b2) &
[b3 + 1,b4] in Indices Gauge(b5,b2) &
(Cage(b5,b2)) /. b1 = (Gauge(b5,b2)) *(b3,b4) &
(Cage(b5,b2)) /. (b1 + 1) = (Gauge(b5,b2)) *(b3 + 1,b4)
holds 1 < b4;
:: JORDAN10:th 4
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st 1 <= b1 &
b1 + 1 <= len Cage(b5,b2) &
[b3,b4] in Indices Gauge(b5,b2) &
[b3 + 1,b4] in Indices Gauge(b5,b2) &
(Cage(b5,b2)) /. b1 = (Gauge(b5,b2)) *(b3 + 1,b4) &
(Cage(b5,b2)) /. (b1 + 1) = (Gauge(b5,b2)) *(b3,b4)
holds b4 < width Gauge(b5,b2);
:: JORDAN10:th 5
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
b2 misses L~ Cage(b2,b1);
:: JORDAN10:th 6
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
N-bound L~ Cage(b2,b1) = (N-bound b2) + (((N-bound b2) - S-bound b2) / (2 |^ b1));
:: JORDAN10:th 7
theorem
for b1, b2 being Element of NAT
for b3 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st b1 < b2
holds N-bound L~ Cage(b3,b2) < N-bound L~ Cage(b3,b1);
:: JORDAN10:funcreg 1
registration
let a1 be connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
let a2 be Element of NAT;
cluster Cl RightComp Cage(a1,a2) -> compact;
end;
:: JORDAN10:th 8
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
N-min b2 in RightComp Cage(b2,b1);
:: JORDAN10:th 9
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
b2 meets RightComp Cage(b2,b1);
:: JORDAN10:th 10
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
b2 misses LeftComp Cage(b2,b1);
:: JORDAN10: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
b2 c= RightComp Cage(b2,b1);
:: JORDAN10: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
b2 c= BDD L~ Cage(b2,b1);
:: JORDAN10: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
UBD L~ Cage(b2,b1) c= UBD b2;
:: JORDAN10:funcnot 1 => JORDAN10:func 1
definition
let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
func UBD-Family A1 -> set equals
{UBD L~ Cage(a1,b1) where b1 is Element of NAT: TRUE};
end;
:: JORDAN10:def 1
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
UBD-Family b1 = {UBD L~ Cage(b1,b2) where b2 is Element of NAT: TRUE};
:: JORDAN10:funcnot 2 => JORDAN10:func 2
definition
let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
func BDD-Family A1 -> set equals
{Cl BDD L~ Cage(a1,b1) where b1 is Element of NAT: TRUE};
end;
:: JORDAN10:def 2
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
BDD-Family b1 = {Cl BDD L~ Cage(b1,b2) where b2 is Element of NAT: TRUE};
:: JORDAN10:funcnot 3 => JORDAN10:func 3
definition
let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
redefine func UBD-Family a1 -> Element of bool bool the carrier of TOP-REAL 2;
end;
:: JORDAN10:funcnot 4 => JORDAN10:func 4
definition
let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
redefine func BDD-Family a1 -> Element of bool bool the carrier of TOP-REAL 2;
end;
:: JORDAN10:funcreg 2
registration
let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
cluster UBD-Family a1 -> non empty;
end;
:: JORDAN10:funcreg 3
registration
let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
cluster BDD-Family a1 -> non empty;
end;
:: JORDAN10:th 14
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
union UBD-Family b1 = UBD b1;
:: JORDAN10:th 15
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
b1 c= meet BDD-Family b1;
:: JORDAN10: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
BDD b2 misses LeftComp Cage(b2,b1);
:: JORDAN10: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
BDD b2 c= RightComp Cage(b2,b1);
:: JORDAN10:th 18
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st b2 is_inside_component_of b3
holds b2 misses L~ Cage(b3,b1);
:: JORDAN10: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
BDD b2 misses L~ Cage(b2,b1);
:: JORDAN10:th 20
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
COMPLEMENT UBD-Family b1 = BDD-Family b1;
:: JORDAN10:th 21
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
meet BDD-Family b1 = b1 \/ BDD b1;