reserve n for Nat;

theorem
  for C be compact connected non vertical non horizontal
  Subset of TOP-REAL 2 holds N-bound C < N-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;
  N-bound C > S-bound C + 0 by SPRECT_1:32;
  then N-bound C - S-bound C > 0 by XREAL_1:20;
  then
A2: (N-bound C - S-bound C)/(2|^n) > N-bound C - N-bound C by A1,XREAL_1:139;
  N-bound L~Cage(C,n) = N-bound C + (N-bound C - S-bound C)/(2|^n)
  by JORDAN10:6;
  hence thesis by A2,XREAL_1:19;
end;
