reserve n for Nat;

theorem
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 holds N-max L~Cage(C,n) in rng Upper_Seq(C,n) & N-max L~Cage(C,n) in
  L~Upper_Seq(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  set x = N-max L~Cage(C,n);
  set p = W-min L~Cage(C,n);
  set f = Rotate(Cage(C,n),E-max L~Cage(C,n));
A1: rng f = rng Cage(C,n) by FINSEQ_6:90,SPRECT_2:46;
A2: x in rng Cage(C,n) by SPRECT_2:40;
A3: Lower_Seq(C,n)/.1 = E-max L~Cage(C,n) by JORDAN1F:6;
A4: L~Cage(C,n) = L~f by REVROT_1:33;
  W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
  then
A5: W-min L~Cage(C,n) in rng f by FINSEQ_6:90,SPRECT_2:46;
A6: p in rng Cage(C,n) by SPRECT_2:43;
  Lower_Seq(C,n) = f-:W-min L~Cage(C,n) by JORDAN1G:18;
  then
A7: Lower_Seq(C,n)/.1 = f/.1 by A5,FINSEQ_5:44;
  then
A8: (W-min L~f)..f < (W-max L~f)..f by A4,JORDAN1F:6,SPRECT_5:42;
A9: (W-max L~f)..f <= (N-min L~f)..f by A7,A4,JORDAN1F:6,SPRECT_5:43;
  per cases;
  suppose
    N-max L~f <> E-max L~f;
    then (W-max L~f)..f < (N-max L~f)..f by A7,A3,A4,A9,SPRECT_5:44,XXREAL_0:2;
    then x in rng(f:-p) by A1,A2,A6,A4,A8,FINSEQ_6:62,XXREAL_0:2;
    hence
A10: x in rng Upper_Seq(C,n) by Th4;
    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 A10;
  end;
  suppose
A11: N-max L~f = E-max L~f;
A12: (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
A13: 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
A14: x in rng Upper_Seq(C,n) by A4,A11,A13,A12,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 A14;
  end;
end;
