reserve n for Nat;

theorem Th30:
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 holds (W-min L~Cage(C,n))..Lower_Seq(C,n) = len Lower_Seq(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
  then
A1: W-min L~Cage(C,n) in rng Rotate(Cage(C,n),E-max L~Cage(C,n)) by FINSEQ_6:90
,SPRECT_2:46;
  Lower_Seq(C,n) = Rotate(Cage(C,n),E-max L~Cage(C,n))-:W-min L~Cage(C,n)
& ( W-min L~Cage(C,n))..Rotate(Cage(C,n),E-max L~Cage(C,n)) <= (W-min L~Cage(C,
  n)) ..Rotate(Cage(C,n),E-max L~Cage(C,n)) by Th18;
  then W-min L~Cage(C,n) in rng Lower_Seq(C,n) by A1,FINSEQ_5:46;
  then
A2: Lower_Seq(C,n) just_once_values W-min L~Cage(C,n) by FINSEQ_4:8;
  Lower_Seq(C,n)/.len Lower_Seq(C,n) = W-min L~Cage(C,n) by JORDAN1F:8;
  hence thesis by A2,FINSEQ_6:166;
end;
