Article JORDAN1E, MML version 4.99.1005

:: JORDAN1E:th 5
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
   st b1 is_in_the_area_of b2
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in proj2 b1
   holds b1 -: b3 is_in_the_area_of b2;

:: JORDAN1E:th 6
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
   st b1 is_in_the_area_of b2
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in proj2 b1
   holds b1 :- b3 is_in_the_area_of b2;

:: JORDAN1E:th 7
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds L_Cut(b1,b2) <> {};

:: JORDAN1E:th 8
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & 2 <= len R_Cut(b1,b2)
   holds b1 . 1 in L~ R_Cut(b1,b2);

:: JORDAN1E:th 9
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
   st b1 is being_S-Seq
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds not b1 . 1 in L~ mid(b1,(Index(b2,b1)) + 1,len b1);

:: JORDAN1E:th 10
theorem
for b1, b2, b3, b4 being Element of NAT
      st b1 + b2 = b3 + b4 & b1 <= b3 & b2 <= b4
   holds b1 = b3;

:: JORDAN1E:th 11
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
   st b1 is being_S-Seq
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & b1 . 1 in L~ L_Cut(b1,b2)
   holds b1 . 1 = b2;

:: JORDAN1E:funcnot 1 => JORDAN1E:func 1
definition
  let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  func Upper_Seq(A1,A2) -> FinSequence of the carrier of TOP-REAL 2 equals
    (Rotate(Cage(a1,a2),W-min L~ Cage(a1,a2))) -: E-max L~ Cage(a1,a2);
end;

:: JORDAN1E:def 1
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   Upper_Seq(b1,b2) = (Rotate(Cage(b1,b2),W-min L~ Cage(b1,b2))) -: E-max L~ Cage(b1,b2);

:: JORDAN1E:th 12
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   len Upper_Seq(b1,b2) = (E-max L~ Cage(b1,b2)) .. Rotate(Cage(b1,b2),W-min L~ Cage(b1,b2));

:: JORDAN1E:funcnot 2 => JORDAN1E:func 2
definition
  let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  func Lower_Seq(A1,A2) -> FinSequence of the carrier of TOP-REAL 2 equals
    (Rotate(Cage(a1,a2),W-min L~ Cage(a1,a2))) :- E-max L~ Cage(a1,a2);
end;

:: JORDAN1E:def 2
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   Lower_Seq(b1,b2) = (Rotate(Cage(b1,b2),W-min L~ Cage(b1,b2))) :- E-max L~ Cage(b1,b2);

:: JORDAN1E:th 13
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   len Lower_Seq(b1,b2) = ((len Rotate(Cage(b1,b2),W-min L~ Cage(b1,b2))) - ((E-max L~ Cage(b1,b2)) .. Rotate(Cage(b1,b2),W-min L~ Cage(b1,b2)))) + 1;

:: JORDAN1E:funcreg 1
registration
  let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Upper_Seq(a1,a2) -> non empty;
end;

:: JORDAN1E:funcreg 2
registration
  let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Lower_Seq(a1,a2) -> non empty;
end;

:: JORDAN1E:funcreg 3
registration
  let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Upper_Seq(a1,a2) -> one-to-one special unfolded s.n.c.;
end;

:: JORDAN1E:funcreg 4
registration
  let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Lower_Seq(a1,a2) -> one-to-one special unfolded s.n.c.;
end;

:: JORDAN1E:th 14
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   (len Upper_Seq(b1,b2)) + len Lower_Seq(b1,b2) = (len Cage(b1,b2)) + 1;

:: JORDAN1E:th 15
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   Rotate(Cage(b1,b2),W-min L~ Cage(b1,b2)) = (Upper_Seq(b1,b2)) ^' Lower_Seq(b1,b2);

:: JORDAN1E:th 16
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   L~ Cage(b1,b2) = L~ ((Upper_Seq(b1,b2)) ^' Lower_Seq(b1,b2));

:: JORDAN1E:th 17
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   L~ Cage(b1,b2) = (L~ Upper_Seq(b1,b2)) \/ L~ Lower_Seq(b1,b2);

:: JORDAN1E:th 18
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   W-min b1 <> E-min b1;

:: JORDAN1E:th 19
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   3 <= len Upper_Seq(b1,b2) & 3 <= len Lower_Seq(b1,b2);

:: JORDAN1E:funcreg 5
registration
  let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Upper_Seq(a1,a2) -> being_S-Seq;
end;

:: JORDAN1E:funcreg 6
registration
  let a1 be compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Lower_Seq(a1,a2) -> being_S-Seq;
end;

:: JORDAN1E:th 20
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   (L~ Upper_Seq(b1,b2)) /\ L~ Lower_Seq(b1,b2) = {W-min L~ Cage(b1,b2),E-max L~ Cage(b1,b2)};

:: JORDAN1E:th 21
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   Upper_Seq(b2,b1) is_in_the_area_of Cage(b2,b1);

:: JORDAN1E:th 22
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   Lower_Seq(b2,b1) is_in_the_area_of Cage(b2,b1);

:: JORDAN1E:th 23
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
   ((Cage(b2,b1)) /. 2) `2 = N-bound L~ Cage(b2,b1);

:: JORDAN1E:th 24
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 1 <= b3 &
         b3 + 1 <= len Cage(b2,b1) &
         (Cage(b2,b1)) /. b3 = E-max L~ Cage(b2,b1)
   holds ((Cage(b2,b1)) /. (b3 + 1)) `1 = E-bound L~ Cage(b2,b1);

:: JORDAN1E:th 25
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 1 <= b3 &
         b3 + 1 <= len Cage(b2,b1) &
         (Cage(b2,b1)) /. b3 = S-max L~ Cage(b2,b1)
   holds ((Cage(b2,b1)) /. (b3 + 1)) `2 = S-bound L~ Cage(b2,b1);

:: JORDAN1E:th 26
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 1 <= b3 &
         b3 + 1 <= len Cage(b2,b1) &
         (Cage(b2,b1)) /. b3 = W-min L~ Cage(b2,b1)
   holds ((Cage(b2,b1)) /. (b3 + 1)) `1 = W-bound L~ Cage(b2,b1);