reserve n for Nat;

theorem Th39:
  for C be compact non vertical non horizontal Subset of TOP-REAL
  2 for n be Nat holds rng Upper_Seq(C,n) c= rng Cage(C,n) & rng
  Lower_Seq(C,n) c= rng Cage(C,n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then
A1: 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),W-min L~Cage(C,n))-:E-max L~Cage(C,n )
  by JORDAN1E:def 1;
  then rng Upper_Seq(C,n) c= rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by
FINSEQ_5:48;
  hence rng Upper_Seq(C,n) c= rng Cage(C,n) by FINSEQ_6:90,SPRECT_2:43;
  Lower_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n)):-E-max L~Cage(C,n )
  by JORDAN1E:def 2;
  then rng Lower_Seq(C,n) c= rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by A1,
FINSEQ_5:55;
  hence thesis by FINSEQ_6:90,SPRECT_2:43;
end;
