reserve n for Nat;

theorem
  for C be compact connected non vertical non horizontal
  Subset of TOP-REAL 2 holds S-bound L~Cage(C,n) < S-bound C
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) > S-bound C - S-bound C by A1,XREAL_1:139;
  S-bound L~Cage(C,n) = S-bound C - (N-bound C - S-bound C)/(2|^n)
  by JORDAN1A:63;
  hence thesis by A2,XREAL_1:11;
end;
