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;