reserve n for Nat;

theorem Th40:
  for C be compact non vertical non horizontal Subset of TOP-REAL
  2 holds Upper_Seq(C,n) is_a_h.c._for Cage(C,n)
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
A1: (Upper_Seq(C,n)/.1)`1 = (W-min L~Cage(C,n))`1 by JORDAN1F:5
    .= W-bound L~Cage(C,n) by EUCLID:52;
A2: (Upper_Seq(C,n)/.len Upper_Seq(C,n))`1 = (E-max L~Cage(C,n))`1 by
JORDAN1F:7
    .= E-bound L~Cage(C,n) by EUCLID:52;
  Upper_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:17;
  hence thesis by A1,A2,SPRECT_2:def 2;
end;
