Article JORDAN8, MML version 4.99.1005

:: JORDAN8:th 1
theorem
for b1 being set
for b2 being FinSequence of b1
      st 2 <= len b2
   holds b2 | 2 = <*b2 /. 1,b2 /. 2*>;

:: JORDAN8:th 2
theorem
for b1 being Element of NAT
for b2 being set
for b3 being FinSequence of b2
      st b1 + 1 <= len b3
   holds b3 | (b1 + 1) = (b3 | b1) ^ <*b3 /. (b1 + 1)*>;

:: JORDAN8:th 3
theorem
for b1 being set
for b2 being tabular FinSequence of b1 * holds
   <*> b1 is_sequence_on b2;

:: JORDAN8:th 5
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being FinSequence of b2
for b4 being tabular FinSequence of b2 *
      st b3 is_sequence_on b4
   holds b3 /^ b1 is_sequence_on b4;

:: JORDAN8:th 6
theorem
for b1 being Element of NAT
for b2 being set
for b3 being FinSequence of b2
for b4 being tabular FinSequence of b2 *
      st 1 <= b1 & b1 + 1 <= len b3 & b3 is_sequence_on b4
   holds ex b5, b6, b7, b8 being Element of NAT st
      [b5,b6] in Indices b4 &
       b3 /. b1 = b4 *(b5,b6) &
       [b7,b8] in Indices b4 &
       b3 /. (b1 + 1) = b4 *(b7,b8) &
       ((b5 = b7 implies b6 + 1 <> b8) & (b5 + 1 = b7 implies b6 <> b8) & (b5 = b7 + 1 implies b6 <> b8) implies b5 = b7 & b6 = b8 + 1);

:: JORDAN8:th 7
theorem
for b1 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b2 being non empty FinSequence of the carrier of TOP-REAL 2
      st b2 is_sequence_on b1
   holds b2 is standard & b2 is special;

:: JORDAN8:th 8
theorem
for b1 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b2 being non empty FinSequence of the carrier of TOP-REAL 2
      st 2 <= len b2 & b2 is_sequence_on b1
   holds b2 is not constant;

:: JORDAN8:th 9
theorem
for b1 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty FinSequence of the carrier of TOP-REAL 2
      st b3 is_sequence_on b1 &
         (ex b4, b5 being Element of NAT st
            [b4,b5] in Indices b1 & b2 = b1 *(b4,b5)) &
         (for b4, b5, b6, b7 being Element of NAT
               st [b4,b5] in Indices b1 & [b6,b7] in Indices b1 & b3 /. len b3 = b1 *(b4,b5) & b2 = b1 *(b6,b7)
            holds (abs (b6 - b4)) + abs (b7 - b5) = 1)
   holds b3 ^ <*b2*> is_sequence_on b1;

:: JORDAN8:th 10
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st b1 + b2 < len b4 &
         1 <= b3 &
         b3 < width b4 &
         cell(b4,b1,b3) meets cell(b4,b1 + b2,b3)
   holds b2 <= 1;

:: JORDAN8:th 11
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
      b1 is vertical
   iff
      E-bound b1 <= W-bound b1;

:: JORDAN8:th 12
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
      b1 is horizontal
   iff
      N-bound b1 <= S-bound b1;

:: JORDAN8:funcnot 1 => JORDAN8:func 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2 be natural set;
  func Gauge(A1,A2) -> tabular FinSequence of (the carrier of TOP-REAL 2) * means
    len it = (2 |^ a2) + 3 &
     len it = width it &
     (for b1, b2 being natural set
           st [b1,b2] in Indices it
        holds it *(b1,b2) = |[(W-bound a1) + ((((E-bound a1) - W-bound a1) / (2 |^ a2)) * (b1 - 2)),(S-bound a1) + ((((N-bound a1) - S-bound a1) / (2 |^ a2)) * (b2 - 2))]|);
end;

:: JORDAN8:def 1
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being natural set
for b3 being tabular FinSequence of (the carrier of TOP-REAL 2) * holds
      b3 = Gauge(b1,b2)
   iff
      len b3 = (2 |^ b2) + 3 &
       len b3 = width b3 &
       (for b4, b5 being natural set
             st [b4,b5] in Indices b3
          holds b3 *(b4,b5) = |[(W-bound b1) + ((((E-bound b1) - W-bound b1) / (2 |^ b2)) * (b4 - 2)),(S-bound b1) + ((((N-bound b1) - S-bound b1) / (2 |^ b2)) * (b5 - 2))]|);

:: JORDAN8:funcreg 1
registration
  let a1 be non empty Element of bool the carrier of TOP-REAL 2;
  let a2 be natural set;
  cluster Gauge(a1,a2) -> non empty-yielding tabular X_equal-in-line Y_equal-in-column;
end;

:: JORDAN8:funcreg 2
registration
  let a1 be non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be natural set;
  cluster Gauge(a1,a2) -> tabular Y_increasing-in-line X_increasing-in-column;
end;

:: JORDAN8:th 13
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being natural set holds
   4 <= len Gauge(b1,b2);

:: JORDAN8:th 14
theorem
for b1, b2 being Element of NAT
for b3 being non empty Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 & b1 <= len Gauge(b3,b2)
   holds ((Gauge(b3,b2)) *(2,b1)) `1 = W-bound b3;

:: JORDAN8:th 15
theorem
for b1, b2 being Element of NAT
for b3 being non empty Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 & b1 <= len Gauge(b3,b2)
   holds ((Gauge(b3,b2)) *((len Gauge(b3,b2)) -' 1,b1)) `1 = E-bound b3;

:: JORDAN8:th 16
theorem
for b1, b2 being Element of NAT
for b3 being non empty Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 & b1 <= len Gauge(b3,b2)
   holds ((Gauge(b3,b2)) *(b1,2)) `2 = S-bound b3;

:: JORDAN8:th 17
theorem
for b1, b2 being Element of NAT
for b3 being non empty Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 & b1 <= len Gauge(b3,b2)
   holds ((Gauge(b3,b2)) *(b1,(len Gauge(b3,b2)) -' 1)) `2 = N-bound b3;

:: JORDAN8:th 18
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being natural set
      st b2 <= len Gauge(b1,b3)
   holds cell(Gauge(b1,b3),b2,len Gauge(b1,b3)) misses b1;

:: JORDAN8:th 19
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being natural set
      st b2 <= len Gauge(b1,b3)
   holds cell(Gauge(b1,b3),len Gauge(b1,b3),b2) misses b1;

:: JORDAN8:th 20
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being natural set
      st b2 <= len Gauge(b1,b3)
   holds cell(Gauge(b1,b3),b2,0) misses b1;

:: JORDAN8:th 21
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being natural set
      st b2 <= len Gauge(b1,b3)
   holds cell(Gauge(b1,b3),0,b2) misses b1;