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) + N-bound L~Cage(C,n) = S-bound C +
  N-bound C
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  thus S-bound L~Cage(C,n)+N-bound L~Cage(C,n) = S-bound L~Cage(C,n)+ (N-bound
  C + (N-bound C - S-bound C)/(2|^n)) by JORDAN10:6
    .= (S-bound C - (N-bound C - S-bound C)/(2|^n))+ (N-bound C + (N-bound C
  - S-bound C)/(2|^n)) by JORDAN1A:63
    .= S-bound C + N-bound C;
end;
