Article JORDAN22, MML version 4.99.1005
:: JORDAN22:th 1
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
(Upper_Appr b1) . b2 c= Cl RightComp Cage(b1,0);
:: JORDAN22:th 2
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
(Lower_Appr b1) . b2 c= Cl RightComp Cage(b1,0);
:: JORDAN22:funcreg 1
registration
let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
cluster Upper_Arc a1 -> non empty connected;
end;
:: JORDAN22:funcreg 2
registration
let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
cluster Lower_Arc a1 -> non empty connected;
end;
:: JORDAN22:th 3
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
(Upper_Appr b1) . b2 is compact(TOP-REAL 2) & (Upper_Appr b1) . b2 is connected(TOP-REAL 2);
:: JORDAN22:th 4
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
(Lower_Appr b1) . b2 is compact(TOP-REAL 2) & (Lower_Appr b1) . b2 is connected(TOP-REAL 2);
:: JORDAN22:funcreg 3
registration
let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
cluster North_Arc a1 -> compact;
end;
:: JORDAN22:funcreg 4
registration
let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
cluster South_Arc a1 -> compact;
end;
:: JORDAN22:th 5
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
[1,1] in Indices Gauge(b1,b2);
:: JORDAN22:th 6
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
[1,2] in Indices Gauge(b1,b2);
:: JORDAN22:th 7
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
[2,1] in Indices Gauge(b1,b2);
:: JORDAN22:th 8
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st b3 < b2 &
[b1,b4] in Indices Gauge(b5,b3) &
[b1,b4 + 1] in Indices Gauge(b5,b3)
holds dist((Gauge(b5,b2)) *(b1,b4),(Gauge(b5,b2)) *(b1,b4 + 1)) < dist((Gauge(b5,b3)) *(b1,b4),(Gauge(b5,b3)) *(b1,b4 + 1));
:: JORDAN22:th 9
theorem
for b1, b2 being Element of NAT
for b3 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st b2 < b1
holds dist((Gauge(b3,b1)) *(1,1),(Gauge(b3,b1)) *(1,2)) < dist((Gauge(b3,b2)) *(1,1),(Gauge(b3,b2)) *(1,2));
:: JORDAN22:th 10
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st b3 < b2 &
[b1,b4] in Indices Gauge(b5,b3) &
[b1 + 1,b4] in Indices Gauge(b5,b3)
holds dist((Gauge(b5,b2)) *(b1,b4),(Gauge(b5,b2)) *(b1 + 1,b4)) < dist((Gauge(b5,b3)) *(b1,b4),(Gauge(b5,b3)) *(b1 + 1,b4));
:: JORDAN22:th 11
theorem
for b1, b2 being Element of NAT
for b3 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st b2 < b1
holds dist((Gauge(b3,b1)) *(1,1),(Gauge(b3,b1)) *(2,1)) < dist((Gauge(b3,b2)) *(1,1),(Gauge(b3,b2)) *(2,1));
:: JORDAN22:th 12
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
for b3, b4 being real set
st 0 < b3 & 0 < b4
holds ex b5 being Element of NAT st
b2 < b5 &
dist((Gauge(b1,b5)) *(1,1),(Gauge(b1,b5)) *(1,2)) < b3 &
dist((Gauge(b1,b5)) *(1,1),(Gauge(b1,b5)) *(2,1)) < b4;
:: JORDAN22:th 13
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds sup (proj2 .: ((L~ Cage(b1,b2)) /\ LSeg((Gauge(b1,b2)) *(Center Gauge(b1,b2),1),(Gauge(b1,b2)) *(Center Gauge(b1,b2),len Gauge(b1,b2))))) = sup (proj2 .: ((L~ Cage(b1,b2)) /\ Vertical_Line (((E-bound L~ Cage(b1,b2)) + W-bound L~ Cage(b1,b2)) / 2)));
:: JORDAN22:th 14
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds inf (proj2 .: ((L~ Cage(b1,b2)) /\ LSeg((Gauge(b1,b2)) *(Center Gauge(b1,b2),1),(Gauge(b1,b2)) *(Center Gauge(b1,b2),len Gauge(b1,b2))))) = inf (proj2 .: ((L~ Cage(b1,b2)) /\ Vertical_Line (((E-bound L~ Cage(b1,b2)) + W-bound L~ Cage(b1,b2)) / 2)));
:: JORDAN22:th 15
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds UMP L~ Cage(b1,b2) = |[((E-bound L~ Cage(b1,b2)) + W-bound L~ Cage(b1,b2)) / 2,sup (proj2 .: ((L~ Cage(b1,b2)) /\ LSeg((Gauge(b1,b2)) *(Center Gauge(b1,b2),1),(Gauge(b1,b2)) *(Center Gauge(b1,b2),len Gauge(b1,b2)))))]|;
:: JORDAN22:th 16
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds LMP L~ Cage(b1,b2) = |[((E-bound L~ Cage(b1,b2)) + W-bound L~ Cage(b1,b2)) / 2,inf (proj2 .: ((L~ Cage(b1,b2)) /\ LSeg((Gauge(b1,b2)) *(Center Gauge(b1,b2),1),(Gauge(b1,b2)) *(Center Gauge(b1,b2),len Gauge(b1,b2)))))]|;
:: JORDAN22:th 17
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
(UMP b1) `2 < (UMP L~ Cage(b1,b2)) `2;
:: JORDAN22:th 18
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
(LMP L~ Cage(b1,b2)) `2 < (LMP b1) `2;
:: JORDAN22:th 21
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds ex b3 being Element of NAT st
1 <= b3 &
b3 <= len Gauge(b1,b2) &
UMP L~ Cage(b1,b2) = (Gauge(b1,b2)) *(Center Gauge(b1,b2),b3);
:: JORDAN22:th 22
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds ex b3 being Element of NAT st
1 <= b3 &
b3 <= len Gauge(b1,b2) &
LMP L~ Cage(b1,b2) = (Gauge(b1,b2)) *(Center Gauge(b1,b2),b3);
:: JORDAN22:th 23
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds UMP L~ Cage(b1,b2) = UMP Upper_Arc L~ Cage(b1,b2);
:: JORDAN22:th 24
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds LMP L~ Cage(b1,b2) = LMP Lower_Arc L~ Cage(b1,b2);
:: JORDAN22:th 25
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds (UMP b1) `2 < (UMP Upper_Arc L~ Cage(b1,b2)) `2;
:: JORDAN22:th 26
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
st 0 < b2
holds (LMP Lower_Arc L~ Cage(b1,b2)) `2 < (LMP b1) `2;
:: JORDAN22:th 27
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 <= b3
holds (UMP L~ Cage(b1,b3)) `2 <= (UMP L~ Cage(b1,b2)) `2;
:: JORDAN22:th 28
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 <= b3
holds (LMP L~ Cage(b1,b2)) `2 <= (LMP L~ Cage(b1,b3)) `2;
:: JORDAN22:th 29
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st 0 < b2 & b2 <= b3
holds (UMP Upper_Arc L~ Cage(b1,b3)) `2 <= (UMP Upper_Arc L~ Cage(b1,b2)) `2;
:: JORDAN22:th 30
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st 0 < b2 & b2 <= b3
holds (LMP Lower_Arc L~ Cage(b1,b2)) `2 <= (LMP Lower_Arc L~ Cage(b1,b3)) `2;
:: JORDAN22:th 31
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
W-min b1 in North_Arc b1;
:: JORDAN22:th 32
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
E-max b1 in North_Arc b1;
:: JORDAN22:th 33
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
W-min b1 in South_Arc b1;
:: JORDAN22:th 34
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
E-max b1 in South_Arc b1;
:: JORDAN22:th 35
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
UMP b1 in North_Arc b1;
:: JORDAN22:th 36
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
LMP b1 in South_Arc b1;
:: JORDAN22:th 37
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
North_Arc b1 c= b1;
:: JORDAN22:th 38
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
South_Arc b1 c= b1;
:: JORDAN22:th 39
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
st (LMP b1 in Lower_Arc b1 implies not UMP b1 in Upper_Arc b1)
holds UMP b1 in Lower_Arc b1 & LMP b1 in Upper_Arc b1;
:: JORDAN22:th 40
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
W-bound b1 = W-bound North_Arc b1;
:: JORDAN22:th 41
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
E-bound b1 = E-bound North_Arc b1;
:: JORDAN22:th 42
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
W-bound b1 = W-bound South_Arc b1;
:: JORDAN22:th 43
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
E-bound b1 = E-bound South_Arc b1;