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 Lower_Seq(C,n) & E-max L~Cage(C,n) in L~
  Lower_Seq(C,n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  set p = E-max L~Cage(C,n);
  Lower_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n)):-p by JORDAN1E:def 2;
  hence
A1: p in rng Lower_Seq(C,n) by FINSEQ_6:61;
  len Lower_Seq(C,n) >= 2 by TOPREAL1:def 8;
  then rng Lower_Seq(C,n) c= L~Lower_Seq(C,n) by SPPOL_2:18;
  hence thesis by A1;
end;
