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 Lower_Seq(C,n) & W-min 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 = W-min L~Cage(C,n);
  Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = p by JORDAN1F:8;
  hence
A1: p in rng Lower_Seq(C,n) by FINSEQ_6:168;
  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;
