reserve n for Nat;

theorem
  for C be compact non vertical non horizontal Subset of TOP-REAL 2
  holds W-min L~Cage(C,n) in rng Upper_Seq(C,n) & W-min 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 p = W-min L~Cage(C,n);
A1: p in rng Cage(C,n) by SPRECT_2:43;
  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),p) 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;
  then Upper_Seq(C,n)/.1 = Rotate(Cage(C,n),p)/.1 by A2,FINSEQ_5:44;
  then Upper_Seq(C,n)/.1 = p by A1,FINSEQ_6:92;
  hence
A3: p in rng Upper_Seq(C,n) by FINSEQ_6:42;
  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;
