reserve C for Simple_closed_curve,
  i, j, n for Nat,
  p for Point of TOP-REAL 2;

theorem Th16:
  for J being Integer st J = [\ ((p`2 - S-bound C) / (N-bound C -
S-bound C) * 2|^n) + 2 /] holds p`2 < (S-bound C) + ((N-bound C - S-bound C)/(2
  |^n))*(J-1)
proof
  set W = S-bound C, E = N-bound C;
  set EW = E - W, PW = p`2 - W;
  let I be Integer;
  set KI = I - 1;
A1: 2|^n > 0 by NEWTON:83;
  assume I = [\ (PW / EW * 2|^n) + 2 /];
  then I > (PW / EW * 2|^n) + 2 - 1 by INT_1:def 6;
  then
A2: I - 1 > (PW / EW * 2|^n) + 1 - 1 by XREAL_1:9;
A3: W + PW = p`2;
A4: EW > 0 by TOPREAL5:16,XREAL_1:50;
  then
A5: EW/(2|^n) > 0 by A1,XREAL_1:139;
  (EW/(2|^n)) * (PW / EW * 2|^n) = PW by A4,A1,Lm2;
  then (EW/(2|^n))*KI > PW by A2,A5,XREAL_1:68;
  hence thesis by A3,XREAL_1:6;
end;
