reserve n for Nat;

theorem Th55:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for n be Nat st n > 0 holds L~Upper_Seq(C,n) = Upper_Arc
  L~Cage(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
A1: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then
A2: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by FINSEQ_6:90
,SPRECT_2:43;
A3: Upper_Seq(C,n)=Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n) by
JORDAN1E:def 1;
  then Upper_Seq(C,n)/.1 = Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by A2,
FINSEQ_5:44;
  then
A4: Upper_Seq(C,n)/.1 = W-min L~Cage(C,n) by A1,FINSEQ_6:92;
  Upper_Seq(C,n)/.len Upper_Seq(C,n) = (Rotate(Cage(C,n),W-min L~Cage(C,n)
)-:E-max L~Cage(C,n))/. ((E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n
  ))) by A3,A2,FINSEQ_5:42
    .= E-max L~Cage(C,n) by A2,FINSEQ_5:45;
  then
A5: L~Upper_Seq(C,n) is_an_arc_of W-min L~Cage(C,n),E-max L~Cage(C,n) by A4,
TOPREAL1:25;
  assume n > 0;
  then
A6: First_Point(L~Upper_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
  Vertical_Line((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))`2 > Last_Point(L~
  Lower_Seq(C,n),E-max L~Cage(C,n),W-min L~Cage(C,n), Vertical_Line((W-bound L~
  Cage(C,n)+E-bound L~Cage(C,n))/2))`2 by Th54;
A7: Lower_Seq(C,n)/.1 = (Rotate(Cage(C,n),W-min L~Cage(C,n)):- E-max L~Cage
  (C,n))/.1 by JORDAN1E:def 2
    .= E-max L~Cage(C,n) by FINSEQ_5:53;
  Lower_Seq(C,n)=Rotate(Cage(C,n),W-min L~Cage(C,n)):-E-max L~Cage(C,n) by
JORDAN1E:def 2;
  then
  Lower_Seq(C,n)/.len Lower_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n)
  )/. (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by A2,FINSEQ_5:54
    .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
    .= W-min L~Cage(C,n) by A1,FINSEQ_6:92;
  then
A8: L~Lower_Seq(C,n) is_an_arc_of E-max L~Cage(C,n),W-min L~Cage(C,n) by A7,
TOPREAL1:25;
  L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) = {W-min L~Cage(C,n),E-max L~Cage(
  C,n)} & L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) = L~Cage(C,n) by JORDAN1E:13,16;
  hence thesis by A5,A8,A6,JORDAN6:def 8;
end;
