reserve n for Nat;

theorem Th7:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 holds N-min L~Cage(C,n) in rng Upper_Seq(C,n) & N-min 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-min 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_2:39;
A3: 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
A4: W-min L~Cage(C,n) in rng f by FINSEQ_6:90,SPRECT_2:46;
A5: 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
A6: Lower_Seq(C,n)/.1 = f/.1 by A4,FINSEQ_5:44;
  then
A7: (W-max L~f)..f <= (N-min L~f)..f by A3,JORDAN1F:6,SPRECT_5:43;
  (W-min L~f)..f < (W-max L~f)..f by A6,A3,JORDAN1F:6,SPRECT_5:42;
  then x in rng(f:-p) by A1,A2,A5,A3,A7,FINSEQ_6:62,XXREAL_0:2;
  hence
A8: 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 A8;
end;
