reserve n for Nat;

theorem Th56:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for n be Nat st n > 0 holds L~Lower_Seq(C,n) = Lower_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;
A2: 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;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then
  Lower_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) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by FINSEQ_6:90
,JORDAN1E:def 2,SPRECT_2:43;
  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 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
A3: L~Lower_Seq(C,n) is_an_arc_of E-max L~Cage(C,n),W-min L~Cage(C,n) by A2,
TOPREAL1:25;
  assume n > 0;
  then
A4: L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) & 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
,Th55;
  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 A3,A4,JORDAN6:def 9;
end;
