reserve n for Nat;

theorem
  for C be compact non vertical non horizontal Subset of TOP-REAL 2
  holds E-max L~Cage(C,n) in rng Upper_Seq(C,n) & E-max L~Cage(C,n) in L~
  Upper_Seq(C,n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  set x = E-max L~Cage(C,n);
  set p = W-min L~Cage(C,n);
A1: (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));
  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;
  Upper_Seq(C,n) = Rotate(Cage(C,n),p)-:E-max L~Cage(C,n) by JORDAN1E:def 1;
  hence
A3: x in rng Upper_Seq(C,n) by A2,A1,FINSEQ_5:46;
  len Upper_Seq(C,n) >= 2 by TOPREAL1:def 8;
  then rng Upper_Seq(C,n) c= L~Upper_Seq(C,n) by SPPOL_2:18;
  hence thesis by A3;
end;
