Article JORDAN1C, MML version 4.99.1005

:: JORDAN1C:th 2
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st [b2,b3] in Indices Gauge(b1,b4) &
         [b2 + 1,b3] in Indices Gauge(b1,b4)
   holds dist((Gauge(b1,b4)) *(1,1),(Gauge(b1,b4)) *(2,1)) = ((Gauge(b1,b4)) *(b2 + 1,b3)) `1 - (((Gauge(b1,b4)) *(b2,b3)) `1);

:: JORDAN1C:th 3
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st [b2,b3] in Indices Gauge(b1,b4) &
         [b2,b3 + 1] in Indices Gauge(b1,b4)
   holds dist((Gauge(b1,b4)) *(1,1),(Gauge(b1,b4)) *(1,2)) = ((Gauge(b1,b4)) *(b2,b3 + 1)) `2 - (((Gauge(b1,b4)) *(b2,b3)) `2);

:: JORDAN1C:th 4
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds proj1 .: b1 is bounded;

:: JORDAN1C:th 5
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2, b3 being non empty Element of bool the carrier of TOP-REAL 2
      st b3 = b1 \/ b2 & proj1 .: b2 is not empty & proj1 .: b2 is bounded_below
   holds W-bound b3 = min(W-bound b1,W-bound b2);

:: JORDAN1C:th 6
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
      st b1 in b2 & b2 is Bounded(2)
   holds W-bound b2 <= b1 `1 & b1 `1 <= E-bound b2 & S-bound b2 <= b1 `2 & b1 `2 <= N-bound b2;

:: JORDAN1C:th 18
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
      st BDD b1 <> {}
   holds W-bound b1 <= W-bound BDD b1;

:: JORDAN1C:th 19
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
      st BDD b1 <> {}
   holds E-bound BDD b1 <= E-bound b1;

:: JORDAN1C:th 20
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
      st BDD b1 <> {}
   holds S-bound b1 <= S-bound BDD b1;

:: JORDAN1C:th 21
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
      st BDD b1 <> {}
   holds N-bound BDD b1 <= N-bound b1;

:: JORDAN1C:th 22
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL 2
for b3 being compact non vertical Element of bool the carrier of TOP-REAL 2
for b4 being integer set
      st b2 in BDD b3 &
         b4 = [\(((b2 `1 - W-bound b3) / ((E-bound b3) - W-bound b3)) * (2 |^ b1)) + 2/]
   holds 1 < b4;

:: JORDAN1C:th 23
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL 2
for b3 being compact non vertical Element of bool the carrier of TOP-REAL 2
for b4 being integer set
      st b2 in BDD b3 &
         b4 = [\(((b2 `1 - W-bound b3) / ((E-bound b3) - W-bound b3)) * (2 |^ b1)) + 2/]
   holds b4 + 1 <= len Gauge(b3,b1);

:: JORDAN1C:th 24
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL 2
for b3 being compact non horizontal Element of bool the carrier of TOP-REAL 2
for b4 being integer set
      st b2 in BDD b3 &
         b4 = [\(((b2 `2 - S-bound b3) / ((N-bound b3) - S-bound b3)) * (2 |^ b1)) + 2/]
   holds 1 < b4 & b4 + 1 <= width Gauge(b3,b1);

:: JORDAN1C: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
for b3 being Element of the carrier of TOP-REAL 2
for b4 being integer set
      st b4 = [\(((b3 `1 - W-bound b1) / ((E-bound b1) - W-bound b1)) * (2 |^ b2)) + 2/]
   holds (W-bound b1) + ((((E-bound b1) - W-bound b1) / (2 |^ b2)) * (b4 - 2)) <= b3 `1;

:: JORDAN1C: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
for b3 being Element of the carrier of TOP-REAL 2
for b4 being integer set
      st b4 = [\(((b3 `1 - W-bound b1) / ((E-bound b1) - W-bound b1)) * (2 |^ b2)) + 2/]
   holds b3 `1 < (W-bound b1) + ((((E-bound b1) - W-bound b1) / (2 |^ b2)) * (b4 - 1));

:: JORDAN1C:th 27
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 being Element of the carrier of TOP-REAL 2
for b4 being integer set
      st b4 = [\(((b3 `2 - S-bound b1) / ((N-bound b1) - S-bound b1)) * (2 |^ b2)) + 2/]
   holds (S-bound b1) + ((((N-bound b1) - S-bound b1) / (2 |^ b2)) * (b4 - 2)) <= b3 `2;

:: JORDAN1C:th 28
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 being Element of the carrier of TOP-REAL 2
for b4 being integer set
      st b4 = [\(((b3 `2 - S-bound b1) / ((N-bound b1) - S-bound b1)) * (2 |^ b2)) + 2/]
   holds b3 `2 < (S-bound b1) + ((((N-bound b1) - S-bound b1) / (2 |^ b2)) * (b4 - 1));

:: JORDAN1C:th 29
theorem
for b1 being closed Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
      st b2 in BDD b1
   holds ex b3 being Element of REAL st
      0 < b3 & Ball(b2,b3) c= BDD b1;

:: JORDAN1C:th 30
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
for b5, b6 being Element of the carrier of TOP-REAL 2
for b7 being real set
      st dist((Gauge(b1,b2)) *(1,1),(Gauge(b1,b2)) *(1,2)) < b7 &
         dist((Gauge(b1,b2)) *(1,1),(Gauge(b1,b2)) *(2,1)) < b7 &
         b5 in cell(Gauge(b1,b2),b3,b4) &
         b6 in cell(Gauge(b1,b2),b3,b4) &
         1 <= b3 &
         b3 + 1 <= len Gauge(b1,b2) &
         1 <= b4 &
         b4 + 1 <= width Gauge(b1,b2)
   holds dist(b5,b6) < 2 * b7;

:: JORDAN1C:th 31
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds b1 `2 <> N-bound BDD b2;

:: JORDAN1C:th 32
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds b1 `1 <> E-bound BDD b2;

:: JORDAN1C:th 33
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds b1 `2 <> S-bound BDD b2;

:: JORDAN1C:th 34
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds b1 `1 <> W-bound BDD b2;

:: JORDAN1C:th 35
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in BDD b1
   holds ex b3, b4, b5 being Element of NAT st
      1 < b4 &
       b4 < len Gauge(b1,b3) &
       1 < b5 &
       b5 < width Gauge(b1,b3) &
       b2 `1 <> ((Gauge(b1,b3)) *(b4,b5)) `1 &
       b2 in cell(Gauge(b1,b3),b4,b5) &
       cell(Gauge(b1,b3),b4,b5) c= BDD b1;

:: JORDAN1C:th 36
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds UBD b1 is not empty;