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

theorem Th15:
  for J being Integer st J = [\ ((p`2 - S-bound C) / (N-bound C -
S-bound C) * 2|^n) + 2 /] holds (S-bound C) + ((N-bound C - S-bound C)/(2|^n))*
  (J-2) <= p`2
proof
  set W = S-bound C, EW = N-bound C - S-bound C;
  set PW = p`2 - W;
  set KI = [\ (PW / EW * 2|^n) /];
  let I be Integer;
A1: EW > 0 by TOPREAL5:16,XREAL_1:50;
  2|^n > 0 by NEWTON:83;
  then
A2: (EW/(2|^n))*(PW / EW * 2|^n) = PW by A1,Lm2;
  assume I = [\ (PW / EW * 2|^n) + 2 /];
  then
A3: I - 2 = [\ (PW / EW * 2|^n) /] by Lm1;
  KI <= PW / EW * 2|^n by INT_1:def 6;
  then
A4: (EW/(2|^n))*KI <= (EW/(2|^n))*(PW / EW * 2|^n) by A1,XREAL_1:64;
  W + PW = p`2;
  hence thesis by A3,A2,A4,XREAL_1:6;
end;
