Article JORDAN14, MML version 4.99.1005

:: JORDAN14:th 1
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st BDD L~ b1 <> RightComp b1
   holds BDD L~ b1 = LeftComp b1;

:: JORDAN14:th 2
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st UBD L~ b1 <> RightComp b1
   holds UBD L~ b1 = LeftComp b1;

:: JORDAN14:th 3
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 FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
      st 1 <= b3 & b3 + 1 <= len b2 & b2 is_sequence_on b1
   holds left_cell(b2,b3,b1) is closed(TOP-REAL 2);

:: JORDAN14:th 4
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, b4 being Element of NAT
      st 1 <= b3 & b3 + 1 <= len b1 & 1 <= b4 & b4 + 1 <= width b1
   holds    b2 in Int cell(b1,b3,b4)
   iff
      (b1 *(b3,b4)) `1 < b2 `1 &
       b2 `1 < (b1 *(b3 + 1,b4)) `1 &
       (b1 *(b3,b4)) `2 < b2 `2 &
       b2 `2 < (b1 *(b3,b4 + 1)) `2;

:: JORDAN14:th 5
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   BDD L~ b1 is connected(TOP-REAL 2);

:: JORDAN14:funcreg 1
registration
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  cluster BDD L~ a1 -> connected;
end;

:: JORDAN14:funcnot 1 => JORDAN14:func 1
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  func SpanStart(A1,A2) -> Element of the carrier of TOP-REAL 2 equals
    (Gauge(a1,a2)) *(X-SpanStart(a1,a2),Y-SpanStart(a1,a2));
end;

:: JORDAN14:def 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
   SpanStart(b1,b2) = (Gauge(b1,b2)) *(X-SpanStart(b1,b2),Y-SpanStart(b1,b2));

:: JORDAN14:th 7
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b2 is_sufficiently_large_for b1
   holds SpanStart(b1,b2) in BDD b1;

:: JORDAN14:th 8
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 is_sufficiently_large_for b1 & 1 <= b3 & b3 + 1 <= len Span(b1,b2)
   holds right_cell(Span(b1,b2),b3,Gauge(b1,b2)) misses b1 &
    left_cell(Span(b1,b2),b3,Gauge(b1,b2)) meets b1;

:: JORDAN14:th 9
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b2 is_sufficiently_large_for b1
   holds b1 misses L~ Span(b1,b2);

:: JORDAN14:funcreg 2
registration
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Cl RightComp Span(a1,a2) -> compact;
end;

:: JORDAN14:th 10
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b2 is_sufficiently_large_for b1
   holds b1 meets LeftComp Span(b1,b2);

:: JORDAN14:th 11
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b2 is_sufficiently_large_for b1
   holds b1 misses RightComp Span(b1,b2);

:: JORDAN14: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
      st b2 is_sufficiently_large_for b1
   holds b1 c= LeftComp Span(b1,b2);

:: JORDAN14: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 b2 is_sufficiently_large_for b1
   holds b1 c= UBD L~ Span(b1,b2);

:: JORDAN14: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 b2 is_sufficiently_large_for b1
   holds BDD L~ Span(b1,b2) c= BDD b1;

:: JORDAN14: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 b2 is_sufficiently_large_for b1
   holds UBD b1 c= UBD L~ Span(b1,b2);

:: JORDAN14: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 b2 is_sufficiently_large_for b1
   holds RightComp Span(b1,b2) c= BDD b1;

:: JORDAN14: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
      st b2 is_sufficiently_large_for b1
   holds UBD b1 c= LeftComp Span(b1,b2);

:: JORDAN14: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
      st b2 is_sufficiently_large_for b1
   holds UBD b1 misses BDD L~ Span(b1,b2);

:: JORDAN14:th 19
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b2 is_sufficiently_large_for b1
   holds UBD b1 misses RightComp Span(b1,b2);

:: JORDAN14:th 20
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st b3 is_sufficiently_large_for b1 & b2 is_outside_component_of b1
   holds b2 misses L~ Span(b1,b3);

:: JORDAN14: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 b2 is_sufficiently_large_for b1
   holds UBD b1 misses L~ Span(b1,b2);

:: JORDAN14: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 b2 is_sufficiently_large_for b1
   holds L~ Span(b1,b2) c= BDD b1;

:: JORDAN14:th 23
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of NAT
      st b5 is_sufficiently_large_for b1 &
         1 <= b4 &
         b4 <= len Span(b1,b5) &
         [b2,b3] in Indices Gauge(b1,b5) &
         (Span(b1,b5)) /. b4 = (Gauge(b1,b5)) *(b2,b3)
   holds 1 < b2;

:: JORDAN14:th 24
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of NAT
      st b5 is_sufficiently_large_for b1 &
         1 <= b4 &
         b4 <= len Span(b1,b5) &
         [b2,b3] in Indices Gauge(b1,b5) &
         (Span(b1,b5)) /. b4 = (Gauge(b1,b5)) *(b2,b3)
   holds b2 < len Gauge(b1,b5);

:: JORDAN14:th 25
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of NAT
      st b5 is_sufficiently_large_for b1 &
         1 <= b4 &
         b4 <= len Span(b1,b5) &
         [b2,b3] in Indices Gauge(b1,b5) &
         (Span(b1,b5)) /. b4 = (Gauge(b1,b5)) *(b2,b3)
   holds 1 < b3;

:: JORDAN14:th 26
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of NAT
      st b5 is_sufficiently_large_for b1 &
         1 <= b4 &
         b4 <= len Span(b1,b5) &
         [b2,b3] in Indices Gauge(b1,b5) &
         (Span(b1,b5)) /. b4 = (Gauge(b1,b5)) *(b2,b3)
   holds b3 < width Gauge(b1,b5);

:: JORDAN14: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
      st b2 is_sufficiently_large_for b1
   holds Y-SpanStart(b1,b2) < width Gauge(b1,b2);

:: JORDAN14:th 28
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st b2 <= b3 & 1 <= b2
   holds X-SpanStart(b1,b3) = ((2 |^ (b3 -' b2)) * ((X-SpanStart(b1,b2)) - 2)) + 2;

:: JORDAN14:th 29
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st b2 <= b3 & b2 is_sufficiently_large_for b1
   holds b3 is_sufficiently_large_for b1;

:: JORDAN14:th 30
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 FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_sequence_on b1 & b2 is special & b3 <= len b1 & b4 <= width b1
   holds (cell(b1,b3,b4)) \ L~ b2 is connected(TOP-REAL 2);

:: JORDAN14:th 31
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 is_sufficiently_large_for b1 &
         Y-SpanStart(b1,b2) <= b3 &
         b3 <= ((2 |^ (b2 -' ApproxIndex b1)) * ((Y-InitStart b1) -' 2)) + 2
   holds (cell(Gauge(b1,b2),(X-SpanStart(b1,b2)) -' 1,b3)) \ L~ Span(b1,b2) c= BDD L~ Span(b1,b2);

:: JORDAN14:th 32
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st b3 <= b2 & 1 < b4 & b4 + 1 < len Gauge(b1,b3)
   holds (((2 |^ (b2 -' b3)) * (b4 - 2)) + 2) + 1 < len Gauge(b1,b2);

:: JORDAN14:th 33
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 is_sufficiently_large_for b1 & b2 <= b3
   holds RightComp Span(b1,b2) meets RightComp Span(b1,b3);

:: JORDAN14:th 34
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 FinSequence of the carrier of TOP-REAL 2
   st b2 is_sequence_on b1 & b2 is special
for b3, b4 being Element of NAT
      st b3 <= len b1 & b4 <= width b1
   holds Int cell(b1,b3,b4) c= (L~ b2) `;

:: JORDAN14:th 35
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 is_sufficiently_large_for b1 & b2 <= b3
   holds L~ Span(b1,b3) c= Cl LeftComp Span(b1,b2);

:: JORDAN14:th 36
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 is_sufficiently_large_for b1 & b2 <= b3
   holds RightComp Span(b1,b2) c= RightComp Span(b1,b3);

:: JORDAN14:th 37
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 is_sufficiently_large_for b1 & b2 <= b3
   holds LeftComp Span(b1,b3) c= LeftComp Span(b1,b2);