Article JORDAN11, MML version 4.99.1005

:: JORDAN11:funcnot 1 => JORDAN11:func 1
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  func ApproxIndex A1 -> Element of NAT means
    it is_sufficiently_large_for a1 &
     (for b1 being Element of NAT
           st b1 is_sufficiently_large_for a1
        holds it <= b1);
end;

:: JORDAN11: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
      b2 = ApproxIndex b1
   iff
      b2 is_sufficiently_large_for b1 &
       (for b3 being Element of NAT
             st b3 is_sufficiently_large_for b1
          holds b2 <= b3);

:: JORDAN11:th 1
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   1 <= ApproxIndex b1;

:: JORDAN11:funcnot 2 => JORDAN11:func 2
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  func Y-InitStart A1 -> Element of NAT means
    it < width Gauge(a1,ApproxIndex a1) &
     cell(Gauge(a1,ApproxIndex a1),(X-SpanStart(a1,ApproxIndex a1)) -' 1,it) c= BDD a1 &
     (for b1 being Element of NAT
           st b1 < width Gauge(a1,ApproxIndex a1) &
              cell(Gauge(a1,ApproxIndex a1),(X-SpanStart(a1,ApproxIndex a1)) -' 1,b1) c= BDD a1
        holds it <= b1);
end;

:: JORDAN11:def 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
      b2 = Y-InitStart b1
   iff
      b2 < width Gauge(b1,ApproxIndex b1) &
       cell(Gauge(b1,ApproxIndex b1),(X-SpanStart(b1,ApproxIndex b1)) -' 1,b2) c= BDD b1 &
       (for b3 being Element of NAT
             st b3 < width Gauge(b1,ApproxIndex b1) &
                cell(Gauge(b1,ApproxIndex b1),(X-SpanStart(b1,ApproxIndex b1)) -' 1,b3) c= BDD b1
          holds b2 <= b3);

:: JORDAN11:th 2
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   1 < Y-InitStart b1;

:: JORDAN11:th 3
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (Y-InitStart b1) + 1 < width Gauge(b1,ApproxIndex b1);

:: JORDAN11:funcnot 3 => JORDAN11:func 3
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  assume a2 is_sufficiently_large_for a1;
  func Y-SpanStart(A1,A2) -> Element of NAT means
    it <= width Gauge(a1,a2) &
     (for b1 being Element of NAT
           st it <= b1 &
              b1 <= ((2 |^ (a2 -' ApproxIndex a1)) * ((Y-InitStart a1) -' 2)) + 2
        holds cell(Gauge(a1,a2),(X-SpanStart(a1,a2)) -' 1,b1) c= BDD a1) &
     (for b1 being Element of NAT
           st b1 <= width Gauge(a1,a2) &
              (for b2 being Element of NAT
                    st b1 <= b2 &
                       b2 <= ((2 |^ (a2 -' ApproxIndex a1)) * ((Y-InitStart a1) -' 2)) + 2
                 holds cell(Gauge(a1,a2),(X-SpanStart(a1,a2)) -' 1,b2) c= BDD a1)
        holds it <= b1);
end;

:: JORDAN11:def 3
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
for b3 being Element of NAT holds
      b3 = Y-SpanStart(b1,b2)
   iff
      b3 <= width Gauge(b1,b2) &
       (for b4 being Element of NAT
             st b3 <= b4 &
                b4 <= ((2 |^ (b2 -' ApproxIndex b1)) * ((Y-InitStart b1) -' 2)) + 2
          holds cell(Gauge(b1,b2),(X-SpanStart(b1,b2)) -' 1,b4) c= BDD b1) &
       (for b4 being Element of NAT
             st b4 <= width Gauge(b1,b2) &
                (for b5 being Element of NAT
                      st b4 <= b5 &
                         b5 <= ((2 |^ (b2 -' ApproxIndex b1)) * ((Y-InitStart b1) -' 2)) + 2
                   holds cell(Gauge(b1,b2),(X-SpanStart(b1,b2)) -' 1,b5) c= BDD b1)
          holds b3 <= b4);

:: JORDAN11:th 4
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 is_sufficiently_large_for b2
   holds X-SpanStart(b2,b1) = ((2 |^ (b1 -' ApproxIndex b2)) * ((X-SpanStart(b2,ApproxIndex b2)) - 2)) + 2;

:: JORDAN11:th 5
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 is_sufficiently_large_for b2
   holds Y-SpanStart(b2,b1) <= ((2 |^ (b1 -' ApproxIndex b2)) * ((Y-InitStart b2) -' 2)) + 2;

:: JORDAN11:th 6
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 is_sufficiently_large_for b2
   holds cell(Gauge(b2,b1),(X-SpanStart(b2,b1)) -' 1,Y-SpanStart(b2,b1)) c= BDD b2;

:: JORDAN11:th 7
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 is_sufficiently_large_for b2
   holds 1 < Y-SpanStart(b2,b1) & Y-SpanStart(b2,b1) <= width Gauge(b2,b1);

:: JORDAN11:th 8
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 is_sufficiently_large_for b2
   holds [X-SpanStart(b2,b1),Y-SpanStart(b2,b1)] in Indices Gauge(b2,b1);

:: JORDAN11:th 9
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 is_sufficiently_large_for b2
   holds [(X-SpanStart(b2,b1)) -' 1,Y-SpanStart(b2,b1)] in Indices Gauge(b2,b1);

:: JORDAN11:th 10
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 is_sufficiently_large_for b2
   holds cell(Gauge(b2,b1),(X-SpanStart(b2,b1)) -' 1,(Y-SpanStart(b2,b1)) -' 1) meets b2;

:: JORDAN11:th 11
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 is_sufficiently_large_for b2
   holds cell(Gauge(b2,b1),(X-SpanStart(b2,b1)) -' 1,Y-SpanStart(b2,b1)) misses b2;

:: JORDAN11:th 12
theorem
for b1, b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
UBD b1 meets UBD b2;

:: JORDAN11:th 13
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 in UBD b1 & b3 in BDD b1
   holds dist(b2,b1) < dist(b2,b3);

:: JORDAN11:funcreg 1
registration
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  cluster BDD a1 -> non empty;
end;

:: JORDAN11:th 14
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 not b2 in BDD b1
   holds dist(b2,b1) <= dist(b2,BDD b1);

:: JORDAN11:th 15
theorem
for b1, b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 c= BDD b2
   holds not b2 c= BDD b1;

:: JORDAN11:th 16
theorem
for b1, b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st b1 c= BDD b2
   holds b2 c= UBD b1;

:: JORDAN11:th 17
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   L~ Cage(b2,b1) c= UBD b2;