Article JORDAN19, MML version 4.99.1005

:: JORDAN19:funcnot 1 => JORDAN19:func 1
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  func Upper_Appr A1 -> Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2 means
    for b1 being Element of NAT holds
       it . b1 = Upper_Arc L~ Cage(a1,b1);
end;

:: JORDAN19:def 1
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2 holds
      b2 = Upper_Appr b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 = Upper_Arc L~ Cage(b1,b3);

:: JORDAN19:funcnot 2 => JORDAN19:func 2
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  func Lower_Appr A1 -> Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2 means
    for b1 being Element of NAT holds
       it . b1 = Lower_Arc L~ Cage(a1,b1);
end;

:: JORDAN19:def 2
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of TOP-REAL 2 holds
      b2 = Lower_Appr b1
   iff
      for b3 being Element of NAT holds
         b2 . b3 = Lower_Arc L~ Cage(b1,b3);

:: JORDAN19:funcnot 3 => JORDAN19:func 3
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  func North_Arc A1 -> Element of bool the carrier of TOP-REAL 2 equals
    Lim_inf Upper_Appr a1;
end;

:: JORDAN19:def 3
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   North_Arc b1 = Lim_inf Upper_Appr b1;

:: JORDAN19:funcnot 4 => JORDAN19:func 4
definition
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  func South_Arc A1 -> Element of bool the carrier of TOP-REAL 2 equals
    Lim_inf Lower_Appr a1;
end;

:: JORDAN19:def 4
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   South_Arc b1 = Lim_inf Lower_Appr b1;

:: JORDAN19:th 1
theorem
for b1, b2 being Element of NAT
      st b1 <= b2 & b1 <> 0
   holds (b2 + 1) / b2 <= (b1 + 1) / b1;

:: JORDAN19:th 2
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st 1 <= b3 & b3 <= b1 & 1 <= b4 & b4 <= width Gauge(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(Center Gauge(b2,b1),width Gauge(b2,b1)),(Gauge(b2,b1)) *(Center Gauge(b2,b1),b4)) c= LSeg((Gauge(b2,b3)) *(Center Gauge(b2,b3),width Gauge(b2,b3)),(Gauge(b2,b1)) *(Center Gauge(b2,b1),b4));

:: JORDAN19:th 3
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 NAT
      st 1 <= b3 &
         b3 <= len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,width Gauge(b2,b1)),(Gauge(b2,b1)) *(b3,b4)) meets L~ Upper_Seq(b2,b1);

:: JORDAN19:th 4
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
   st 0 < b2
for b3, b4 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b1,b2) &
         1 <= b4 &
         b4 <= width Gauge(b1,b2) &
         (Gauge(b1,b2)) *(b3,b4) in L~ Cage(b1,b2)
   holds LSeg((Gauge(b1,b2)) *(b3,width Gauge(b1,b2)),(Gauge(b1,b2)) *(b3,b4)) meets Upper_Arc L~ Cage(b1,b2);

:: JORDAN19:th 5
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 NAT
      st (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3) in Lower_Arc L~ Cage(b2,b1 + 1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1 + 1)
   holds LSeg((Gauge(b2,1)) *(Center Gauge(b2,1),width Gauge(b2,1)),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3)) meets Upper_Arc L~ Cage(b2,b1 + 1);

:: JORDAN19:th 6
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 FinSequence of the carrier of TOP-REAL 2
for b4 being Element of NAT
      st 1 <= b4 &
         b4 + 1 <= len b3 &
         b3 is_sequence_on Gauge(b2,b1) &
         dist(b3 /. b4,b3 /. (b4 + 1)) <> ((N-bound b2) - S-bound b2) / (2 |^ b1)
   holds dist(b3 /. b4,b3 /. (b4 + 1)) = ((E-bound b2) - W-bound b2) / (2 |^ b1);

:: JORDAN19:th 7
theorem
for b1 being symmetric triangle MetrStruct
for b2 being real set
for b3, b4, b5 being Element of the carrier of b1
      st b3 in Ball(b5,b2) & b4 in Ball(b5,b2)
   holds dist(b3,b4) < 2 * b2;

:: JORDAN19:th 9
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
   N-bound b2 < N-bound L~ Cage(b2,b1);

:: JORDAN19:th 10
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 b2 < E-bound L~ Cage(b2,b1);

:: JORDAN19:th 11
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;

:: JORDAN19:th 12
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;

:: JORDAN19:th 13
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b4 &
         b4 <= width Gauge(b2,b1) &
         (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b4))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b4))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b4)}
   holds LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b4)) meets Upper_Arc b2;

:: JORDAN19:th 14
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b4 &
         b4 <= width Gauge(b2,b1) &
         (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b4))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b4))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b4)}
   holds LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b4)) meets Lower_Arc b2;

:: JORDAN19:th 15
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         0 < b1 &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ Lower_Arc L~ Cage(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ Upper_Arc L~ Cage(b2,b1) = {(Gauge(b2,b1)) *(b3,b4)}
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Upper_Arc b2;

:: JORDAN19:th 16
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         0 < b1 &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ Lower_Arc L~ Cage(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ Upper_Arc L~ Cage(b2,b1) = {(Gauge(b2,b1)) *(b3,b4)}
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Lower_Arc b2;

:: JORDAN19: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
for b3, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b5) in L~ Lower_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Upper_Seq(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Upper_Arc b2;

:: JORDAN19:th 18
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b5) in L~ Lower_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Upper_Seq(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Lower_Arc b2;

:: JORDAN19:th 19
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         0 < b1 &
         (Gauge(b2,b1)) *(b3,b5) in Lower_Arc L~ Cage(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in Upper_Arc L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Upper_Arc b2;

:: JORDAN19:th 20
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         0 < b1 &
         (Gauge(b2,b1)) *(b3,b5) in Lower_Arc L~ Cage(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in Upper_Arc L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Lower_Arc b2;

:: JORDAN19:th 21
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of NAT
      st 1 < b3 &
         b3 <= b4 &
         b4 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1) &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b6)}
   holds (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6)) meets Upper_Arc b2;

:: JORDAN19:th 22
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of NAT
      st 1 < b3 &
         b3 <= b4 &
         b4 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1) &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b6)}
   holds (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6)) meets Lower_Arc b2;

:: JORDAN19:th 23
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of NAT
      st 1 < b4 &
         b4 <= b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1) &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b6)}
   holds (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6)) meets Upper_Arc b2;

:: JORDAN19:th 24
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of NAT
      st 1 < b4 &
         b4 <= b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1) &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b6)}
   holds (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6)) meets Lower_Arc b2;

:: JORDAN19:th 25
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1 + 1) &
         1 < b4 &
         b4 < len Gauge(b2,b1 + 1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,b6) in Lower_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b4,b5) in Upper_Arc L~ Cage(b2,b1 + 1)
   holds (LSeg((Gauge(b2,b1 + 1)) *(b4,b5),(Gauge(b2,b1 + 1)) *(b4,b6))) \/ LSeg((Gauge(b2,b1 + 1)) *(b4,b6),(Gauge(b2,b1 + 1)) *(b3,b6)) meets Lower_Arc b2;

:: JORDAN19:th 26
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1 + 1) &
         1 < b4 &
         b4 < len Gauge(b2,b1 + 1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,b6) in Lower_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b4,b5) in Upper_Arc L~ Cage(b2,b1 + 1)
   holds (LSeg((Gauge(b2,b1 + 1)) *(b4,b5),(Gauge(b2,b1 + 1)) *(b4,b6))) \/ LSeg((Gauge(b2,b1 + 1)) *(b4,b6),(Gauge(b2,b1 + 1)) *(b3,b6)) meets Upper_Arc b2;

:: JORDAN19:th 27
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 W-bound b1 < b2 `1 & b2 `1 < E-bound b1 & b2 in North_Arc b1
   holds not b2 in South_Arc b1;

:: JORDAN19:th 28
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 `1 = ((W-bound b1) + E-bound b1) / 2 &
         b2 in North_Arc b1
   holds not b2 in South_Arc b1;