reserve n for Nat;

theorem Th24:
  for C be compact non vertical non horizontal Subset of TOP-REAL
  2 holds (E-max L~Cage(C,n))..Upper_Seq(C,n) = len Upper_Seq(C,n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then
A1: 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;
  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)) <= (E-max L~Cage(C,
  n)) ..Rotate(Cage(C,n),W-min L~Cage(C,n)) by JORDAN1E:def 1;
  then E-max L~Cage(C,n) in rng Upper_Seq(C,n) by A1,FINSEQ_5:46;
  then
A2: Upper_Seq(C,n) just_once_values E-max L~Cage(C,n) by FINSEQ_4:8;
  Upper_Seq(C,n)/.len Upper_Seq(C,n) = E-max L~Cage(C,n) by JORDAN1F:7;
  hence thesis by A2,FINSEQ_6:166;
end;
