Article JORDAN1A, MML version 4.99.1005

:: JORDAN1A:funcnot 1 => JORDAN1A:func 1
definition
  let a1 be Relation-like Function-like FinSequence-like set;
  func Center A1 -> Element of NAT equals
    ((len a1) div 2) + 1;
end;

:: JORDAN1A:def 1
theorem
for b1 being Relation-like Function-like FinSequence-like set holds
   Center b1 = ((len b1) div 2) + 1;

:: JORDAN1A:th 9
theorem
for b1 being Relation-like Function-like FinSequence-like set
      st len b1 is odd
   holds len b1 = (2 * Center b1) - 1;

:: JORDAN1A:th 10
theorem
for b1 being Relation-like Function-like FinSequence-like set
      st len b1 is even
   holds len b1 = (2 * Center b1) - 2;

:: JORDAN1A:exreg 1
registration
  cluster non empty being_simple_closed_curve compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
end;

:: JORDAN1A:th 11
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in N-most b1
   holds b2 `2 = N-bound b1;

:: JORDAN1A:th 12
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in E-most b1
   holds b2 `1 = E-bound b1;

:: JORDAN1A:th 13
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in S-most b1
   holds b2 `2 = S-bound b1;

:: JORDAN1A:th 14
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in W-most b1
   holds b2 `1 = W-bound b1;

:: JORDAN1A:th 15
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   BDD b1 misses b1;

:: JORDAN1A:th 17
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   b1 in Vertical_Line (b1 `1);

:: JORDAN1A:th 18
theorem
for b1, b2 being real set holds
|[b1,b2]| in Vertical_Line b1;

:: JORDAN1A:th 19
theorem
for b1 being real set
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 c= Vertical_Line b1
   holds b2 is vertical;

:: JORDAN1A:th 21
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being real set
      st b1 `1 = b2 `1 &
         b3 in [.proj2 . b1,proj2 . b2.]
   holds |[b1 `1,b3]| in LSeg(b1,b2);

:: JORDAN1A:th 22
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being real set
      st b1 `2 = b2 `2 &
         b3 in [.proj1 . b1,proj1 . b2.]
   holds |[b3,b1 `2]| in LSeg(b1,b2);

:: JORDAN1A:th 23
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being real set
      st b1 in Vertical_Line b3 & b2 in Vertical_Line b3
   holds LSeg(b1,b2) c= Vertical_Line b3;

:: JORDAN1A:th 24
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 meets b2
   holds proj2 .: b1 meets proj2 .: b2;

:: JORDAN1A:th 25
theorem
for b1 being real set
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b2 misses b3 & b2 c= Vertical_Line b1 & b3 c= Vertical_Line b1
   holds proj2 .: b2 misses proj2 .: b3;

:: JORDAN1A:th 37
theorem
for b1, b2 being Element of NAT
for b3 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 1 <= b1 & b1 <= len b3 & 1 <= b2 & b2 <= width b3
   holds b3 *(b1,b2) in LSeg(b3 *(b1,1),b3 *(b1,width b3));

:: JORDAN1A:th 38
theorem
for b1, b2 being Element of NAT
for b3 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 1 <= b1 & b1 <= len b3 & 1 <= b2 & b2 <= width b3
   holds b3 *(b1,b2) in LSeg(b3 *(1,b2),b3 *(len b3,b2));

:: JORDAN1A:th 39
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 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 1 <= b1 & b1 <= width b5 & 1 <= b2 & b2 <= width b5 & 1 <= b3 & b3 <= b4 & b4 <= len b5
   holds (b5 *(b3,b1)) `1 <= (b5 *(b4,b2)) `1;

:: JORDAN1A:th 40
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 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 1 <= b1 & b1 <= len b5 & 1 <= b2 & b2 <= len b5 & 1 <= b3 & b3 <= b4 & b4 <= width b5
   holds (b5 *(b1,b3)) `2 <= (b5 *(b2,b4)) `2;

:: JORDAN1A:th 41
theorem
for b1 being Element of NAT
for b2 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 b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st b3 is_sequence_on b2 & 1 <= b1 & b1 <= len b2
   holds N-bound L~ b3 <= (b2 *(b1,width b2)) `2;

:: JORDAN1A:th 42
theorem
for b1 being Element of NAT
for b2 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 b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st b3 is_sequence_on b2 & 1 <= b1 & b1 <= width b2
   holds (b2 *(1,b1)) `1 <= W-bound L~ b3;

:: JORDAN1A:th 43
theorem
for b1 being Element of NAT
for b2 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 b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st b3 is_sequence_on b2 & 1 <= b1 & b1 <= len b2
   holds (b2 *(b1,1)) `2 <= S-bound L~ b3;

:: JORDAN1A:th 44
theorem
for b1 being Element of NAT
for b2 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 b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st b3 is_sequence_on b2 & 1 <= b1 & b1 <= width b2
   holds E-bound L~ b3 <= (b2 *(len b2,b1)) `1;

:: JORDAN1A:th 45
theorem
for b1, b2 being Element of NAT
for b3 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 <= len b3 & b2 <= width b3
   holds cell(b3,b1,b2) is not empty;

:: JORDAN1A:th 46
theorem
for b1, b2 being Element of NAT
for b3 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 <= len b3 & b2 <= width b3
   holds cell(b3,b1,b2) is connected(TOP-REAL 2);

:: JORDAN1A:th 47
theorem
for b1 being Element of NAT
for b2 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 <= len b2
   holds cell(b2,b1,0) is not Bounded(2);

:: JORDAN1A:th 48
theorem
for b1 being Element of NAT
for b2 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 <= len b2
   holds cell(b2,b1,width b2) is not Bounded(2);

:: JORDAN1A:th 49
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL 2 holds
   width Gauge(b2,b1) = (2 |^ b1) + 3;

:: JORDAN1A:th 50
theorem
for b1, b2 being Element of NAT
for b3 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 < b2
   holds len Gauge(b3,b1) < len Gauge(b3,b2);

:: JORDAN1A:th 51
theorem
for b1, b2 being Element of NAT
for b3 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 <= b2
   holds len Gauge(b3,b1) <= len Gauge(b3,b2);

:: JORDAN1A:th 52
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 <= b2 & 1 < b3 & b3 < len Gauge(b4,b1)
   holds 1 < ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 &
    ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 < len Gauge(b4,b2);

:: JORDAN1A:th 53
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 <= b2 & 1 < b3 & b3 < width Gauge(b4,b1)
   holds 1 < ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 &
    ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 < width Gauge(b4,b2);

:: JORDAN1A:th 54
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non empty Element of bool the carrier of TOP-REAL 2
   st b1 <= b2 & 1 < b3 & b3 < len Gauge(b5,b1) & 1 < b4 & b4 < width Gauge(b5,b1)
for b6, b7 being Element of NAT
      st b6 = ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 &
         b7 = ((2 |^ (b2 -' b1)) * (b4 - 2)) + 2
   holds (Gauge(b5,b1)) *(b3,b4) = (Gauge(b5,b2)) *(b6,b7);

:: JORDAN1A:th 55
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 <= b2 & 1 < b3 & b3 + 1 < len Gauge(b4,b1)
   holds 1 < ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 &
    ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 <= len Gauge(b4,b2);

:: JORDAN1A:th 56
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty Element of bool the carrier of TOP-REAL 2
      st b1 <= b2 & 1 < b3 & b3 + 1 < width Gauge(b4,b1)
   holds 1 < ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 &
    ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 <= width Gauge(b4,b2);

:: JORDAN1A:th 57
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non empty Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len Gauge(b5,b2) &
         1 <= b3 &
         b3 <= len Gauge(b5,b4) &
         (0 < b2 & 0 < b4 or b2 = 0 & b4 = 0)
   holds ((Gauge(b5,b2)) *(Center Gauge(b5,b2),b1)) `1 = ((Gauge(b5,b4)) *(Center Gauge(b5,b4),b3)) `1;

:: JORDAN1A:th 58
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non empty Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len Gauge(b5,b2) &
         1 <= b3 &
         b3 <= len Gauge(b5,b4) &
         (0 < b2 & 0 < b4 or b2 = 0 & b4 = 0)
   holds ((Gauge(b5,b2)) *(b1,Center Gauge(b5,b2))) `2 = ((Gauge(b5,b4)) *(b3,Center Gauge(b5,b4))) `2;

:: JORDAN1A:th 59
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
      st 1 <= b1 & b1 <= len Gauge(b2,1)
   holds ((Gauge(b2,1)) *(Center Gauge(b2,1),b1)) `1 = ((W-bound b2) + E-bound b2) / 2;

:: JORDAN1A:th 60
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
      st 1 <= b1 & b1 <= len Gauge(b2,1)
   holds ((Gauge(b2,1)) *(b1,Center Gauge(b2,1))) `2 = ((S-bound b2) + N-bound b2) / 2;

:: JORDAN1A:th 61
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 1 <= b1 & b1 <= len Gauge(b5,b2) & 1 <= b3 & b3 <= len Gauge(b5,b4) & b4 <= b2
   holds ((Gauge(b5,b2)) *(b1,len Gauge(b5,b2))) `2 <= ((Gauge(b5,b4)) *(b3,len Gauge(b5,b4))) `2;

:: JORDAN1A:th 62
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 1 <= b1 & b1 <= len Gauge(b5,b2) & 1 <= b3 & b3 <= len Gauge(b5,b4) & b4 <= b2
   holds ((Gauge(b5,b2)) *(len Gauge(b5,b2),b1)) `1 <= ((Gauge(b5,b4)) *(len Gauge(b5,b4),b3)) `1;

:: JORDAN1A:th 63
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 1 <= b1 & b1 <= len Gauge(b5,b2) & 1 <= b3 & b3 <= len Gauge(b5,b4) & b4 <= b2
   holds ((Gauge(b5,b4)) *(1,b3)) `1 <= ((Gauge(b5,b2)) *(1,b1)) `1;

:: JORDAN1A:th 64
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 1 <= b1 & b1 <= len Gauge(b5,b2) & 1 <= b3 & b3 <= len Gauge(b5,b4) & b4 <= b2
   holds ((Gauge(b5,b4)) *(b3,1)) `2 <= ((Gauge(b5,b2)) *(b1,1)) `2;

:: JORDAN1A:th 65
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 1 <= b1 & b1 <= b2
   holds LSeg((Gauge(b3,b2)) *(Center Gauge(b3,b2),1),(Gauge(b3,b2)) *(Center Gauge(b3,b2),len Gauge(b3,b2))) c= LSeg((Gauge(b3,b1)) *(Center Gauge(b3,b1),1),(Gauge(b3,b1)) *(Center Gauge(b3,b1),len Gauge(b3,b1)));

:: JORDAN1A:th 66
theorem
for b1, b2, b3 being Element of NAT
for b4 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 & b1 <= b2 & 1 <= b3 & b3 <= width Gauge(b4,b2)
   holds LSeg((Gauge(b4,b2)) *(Center Gauge(b4,b2),1),(Gauge(b4,b2)) *(Center Gauge(b4,b2),b3)) c= LSeg((Gauge(b4,b1)) *(Center Gauge(b4,b1),1),(Gauge(b4,b2)) *(Center Gauge(b4,b2),b3));

:: JORDAN1A:th 67
theorem
for b1, b2, b3 being Element of NAT
for b4 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 & b1 <= b2 & 1 <= b3 & b3 <= width Gauge(b4,b2)
   holds LSeg((Gauge(b4,b1)) *(Center Gauge(b4,b1),1),(Gauge(b4,b2)) *(Center Gauge(b4,b2),b3)) c= LSeg((Gauge(b4,b1)) *(Center Gauge(b4,b1),1),(Gauge(b4,b1)) *(Center Gauge(b4,b1),len Gauge(b4,b1)));

:: JORDAN1A:th 68
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 b1 <= b2 & 1 < b3 & b3 + 1 < len Gauge(b5,b1) & 1 < b4 & b4 + 1 < width Gauge(b5,b1)
for b6, b7 being Element of NAT
      st ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 <= b6 &
         b6 < ((2 |^ (b2 -' b1)) * (b3 - 1)) + 2 &
         ((2 |^ (b2 -' b1)) * (b4 - 2)) + 2 <= b7 &
         b7 < ((2 |^ (b2 -' b1)) * (b4 - 1)) + 2
   holds cell(Gauge(b5,b2),b6,b7) c= cell(Gauge(b5,b1),b3,b4);

:: JORDAN1A:th 69
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 b1 <= b2 & 3 <= b3 & b3 < len Gauge(b5,b1) & 1 < b4 & b4 + 1 < width Gauge(b5,b1)
for b6, b7 being Element of NAT
      st b6 = ((2 |^ (b2 -' b1)) * (b3 - 2)) + 2 &
         b7 = ((2 |^ (b2 -' b1)) * (b4 - 2)) + 2
   holds cell(Gauge(b5,b2),b6 -' 1,b7) c= cell(Gauge(b5,b1),b3 -' 1,b4);

:: JORDAN1A:th 70
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 b1 <= len Gauge(b3,b2)
   holds cell(Gauge(b3,b2),b1,0) c= UBD b3;

:: JORDAN1A:th 71
theorem
for b1, b2 being Element of NAT
for b3, b4 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st b1 <= len Gauge(b3,b2)
   holds cell(Gauge(b3,b2),b1,width Gauge(b3,b2)) c= UBD b3;

:: JORDAN1A:th 72
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in b2
   holds north_halfline b3 meets L~ Cage(b2,b1);

:: JORDAN1A:th 73
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in b2
   holds east_halfline b3 meets L~ Cage(b2,b1);

:: JORDAN1A:th 74
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in b2
   holds south_halfline b3 meets L~ Cage(b2,b1);

:: JORDAN1A:th 75
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in b2
   holds west_halfline b3 meets L~ Cage(b2,b1);

:: JORDAN1A:th 76
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
   ex b3, b4 being Element of NAT st
      1 <= b3 &
       b3 < len Cage(b2,b1) &
       1 <= b4 &
       b4 <= width Gauge(b2,b1) &
       (Cage(b2,b1)) /. b3 = (Gauge(b2,b1)) *(1,b4);

:: JORDAN1A:th 77
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
   ex b3, b4 being Element of NAT st
      1 <= b3 &
       b3 < len Cage(b2,b1) &
       1 <= b4 &
       b4 <= len Gauge(b2,b1) &
       (Cage(b2,b1)) /. b3 = (Gauge(b2,b1)) *(b4,1);

:: JORDAN1A:th 78
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
   ex b3, b4 being Element of NAT st
      1 <= b3 &
       b3 < len Cage(b2,b1) &
       1 <= b4 &
       b4 <= width Gauge(b2,b1) &
       (Cage(b2,b1)) /. b3 = (Gauge(b2,b1)) *(len Gauge(b2,b1),b4);

:: JORDAN1A:th 79
theorem
for b1, b2, b3 being Element of NAT
for b4 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len Cage(b4,b2) &
         1 <= b3 &
         b3 <= len Gauge(b4,b2) &
         (Cage(b4,b2)) /. b1 = (Gauge(b4,b2)) *(b3,width Gauge(b4,b2))
   holds (Cage(b4,b2)) /. b1 in N-most L~ Cage(b4,b2);

:: JORDAN1A:th 80
theorem
for b1, b2, b3 being Element of NAT
for b4 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len Cage(b4,b2) &
         1 <= b3 &
         b3 <= width Gauge(b4,b2) &
         (Cage(b4,b2)) /. b1 = (Gauge(b4,b2)) *(1,b3)
   holds (Cage(b4,b2)) /. b1 in W-most L~ Cage(b4,b2);

:: JORDAN1A:th 81
theorem
for b1, b2, b3 being Element of NAT
for b4 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len Cage(b4,b2) &
         1 <= b3 &
         b3 <= len Gauge(b4,b2) &
         (Cage(b4,b2)) /. b1 = (Gauge(b4,b2)) *(b3,1)
   holds (Cage(b4,b2)) /. b1 in S-most L~ Cage(b4,b2);

:: JORDAN1A:th 82
theorem
for b1, b2, b3 being Element of NAT
for b4 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len Cage(b4,b2) &
         1 <= b3 &
         b3 <= width Gauge(b4,b2) &
         (Cage(b4,b2)) /. b1 = (Gauge(b4,b2)) *(len Gauge(b4,b2),b3)
   holds (Cage(b4,b2)) /. b1 in E-most L~ Cage(b4,b2);

:: JORDAN1A:th 83
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
   W-bound L~ Cage(b2,b1) = (W-bound b2) - (((E-bound b2) - W-bound b2) / (2 |^ b1));

:: JORDAN1A:th 84
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
   S-bound L~ Cage(b2,b1) = (S-bound b2) - (((N-bound b2) - S-bound b2) / (2 |^ b1));

:: JORDAN1A:th 85
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
   E-bound L~ Cage(b2,b1) = (E-bound b2) + (((E-bound b2) - W-bound b2) / (2 |^ b1));

:: JORDAN1A:th 86
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 holds
   (N-bound L~ Cage(b3,b1)) + S-bound L~ Cage(b3,b1) = (N-bound L~ Cage(b3,b2)) + S-bound L~ Cage(b3,b2);

:: JORDAN1A:th 87
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 holds
   (E-bound L~ Cage(b3,b1)) + W-bound L~ Cage(b3,b1) = (E-bound L~ Cage(b3,b2)) + W-bound L~ Cage(b3,b2);

:: JORDAN1A:th 88
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 E-bound L~ Cage(b3,b2) < E-bound L~ Cage(b3,b1);

:: JORDAN1A:th 89
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 W-bound L~ Cage(b3,b1) < W-bound L~ Cage(b3,b2);

:: JORDAN1A:th 90
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 S-bound L~ Cage(b3,b1) < S-bound L~ Cage(b3,b2);

:: JORDAN1A:th 91
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 1 <= b1 & b1 <= len Gauge(b3,b2)
   holds N-bound L~ Cage(b3,b2) = ((Gauge(b3,b2)) *(b1,len Gauge(b3,b2))) `2;

:: JORDAN1A:th 92
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 1 <= b1 & b1 <= len Gauge(b3,b2)
   holds E-bound L~ Cage(b3,b2) = ((Gauge(b3,b2)) *(len Gauge(b3,b2),b1)) `1;

:: JORDAN1A:th 93
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 1 <= b1 & b1 <= len Gauge(b3,b2)
   holds S-bound L~ Cage(b3,b2) = ((Gauge(b3,b2)) *(b1,1)) `2;

:: JORDAN1A:th 94
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 1 <= b1 & b1 <= len Gauge(b3,b2)
   holds W-bound L~ Cage(b3,b2) = ((Gauge(b3,b2)) *(1,b1)) `1;

:: JORDAN1A:th 95
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
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in b2 &
         b4 in (north_halfline b3) /\ L~ Cage(b2,b1)
   holds b3 `2 < b4 `2;

:: JORDAN1A:th 96
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
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in b2 & b4 in (east_halfline b3) /\ L~ Cage(b2,b1)
   holds b3 `1 < b4 `1;

:: JORDAN1A:th 97
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
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in b2 &
         b4 in (south_halfline b3) /\ L~ Cage(b2,b1)
   holds b4 `2 < b3 `2;

:: JORDAN1A:th 98
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
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in b2 & b4 in (west_halfline b3) /\ L~ Cage(b2,b1)
   holds b4 `1 < b3 `1;

:: JORDAN1A:th 99
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
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 in N-most b3 & b5 in north_halfline b4 & 1 <= b1 & b1 < len Cage(b3,b2) & b5 in LSeg(Cage(b3,b2),b1)
   holds LSeg(Cage(b3,b2),b1) is horizontal;

:: JORDAN1A:th 100
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
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 in E-most b3 & b5 in east_halfline b4 & 1 <= b1 & b1 < len Cage(b3,b2) & b5 in LSeg(Cage(b3,b2),b1)
   holds LSeg(Cage(b3,b2),b1) is vertical;

:: JORDAN1A:th 101
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
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 in S-most b3 & b5 in south_halfline b4 & 1 <= b1 & b1 < len Cage(b3,b2) & b5 in LSeg(Cage(b3,b2),b1)
   holds LSeg(Cage(b3,b2),b1) is horizontal;

:: JORDAN1A:th 102
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
for b4, b5 being Element of the carrier of TOP-REAL 2
      st b4 in W-most b3 & b5 in west_halfline b4 & 1 <= b1 & b1 < len Cage(b3,b2) & b5 in LSeg(Cage(b3,b2),b1)
   holds LSeg(Cage(b3,b2),b1) is vertical;

:: JORDAN1A:th 103
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
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in N-most b2 &
         b4 in (north_halfline b3) /\ L~ Cage(b2,b1)
   holds b4 `2 = N-bound L~ Cage(b2,b1);

:: JORDAN1A:th 104
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
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in E-most b2 & b4 in (east_halfline b3) /\ L~ Cage(b2,b1)
   holds b4 `1 = E-bound L~ Cage(b2,b1);

:: JORDAN1A:th 105
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
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in S-most b2 &
         b4 in (south_halfline b3) /\ L~ Cage(b2,b1)
   holds b4 `2 = S-bound L~ Cage(b2,b1);

:: JORDAN1A:th 106
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
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in W-most b2 & b4 in (west_halfline b3) /\ L~ Cage(b2,b1)
   holds b4 `1 = W-bound L~ Cage(b2,b1);

:: JORDAN1A:th 107
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in N-most b2
   holds ex b4 being Element of the carrier of TOP-REAL 2 st
      (north_halfline b3) /\ L~ Cage(b2,b1) = {b4};

:: JORDAN1A:th 108
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in E-most b2
   holds ex b4 being Element of the carrier of TOP-REAL 2 st
      (east_halfline b3) /\ L~ Cage(b2,b1) = {b4};

:: JORDAN1A:th 109
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in S-most b2
   holds ex b4 being Element of the carrier of TOP-REAL 2 st
      (south_halfline b3) /\ L~ Cage(b2,b1) = {b4};

:: JORDAN1A:th 110
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
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in W-most b2
   holds ex b4 being Element of the carrier of TOP-REAL 2 st
      (west_halfline b3) /\ L~ Cage(b2,b1) = {b4};