reserve n for Nat;

theorem Th9:
  for C be compact connected non vertical non horizontal
  Subset of TOP-REAL 2 holds E-bound C < E-bound L~Cage(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
A1: 2|^n > 0 by NEWTON:83;
  E-bound C > W-bound C + 0 by SPRECT_1:31;
  then E-bound C - W-bound C > 0 by XREAL_1:20;
  then
A2: (E-bound C - W-bound C)/(2|^n) > E-bound C - E-bound C by A1,XREAL_1:139;
  E-bound L~Cage(C,n) = E-bound C + (E-bound C - W-bound C)/(2|^n)
  by JORDAN1A:64;
  hence thesis by A2,XREAL_1:19;
end;
