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

theorem Th13:
  for I being Integer st I = [\ ((p`1 - W-bound C) / (E-bound C -
W-bound C) * 2|^n) + 2 /] holds (W-bound C) + (((E-bound C)-(W-bound C))/(2|^n)
  )*(I-2) <= p`1
proof
  set W = W-bound C, EW = E-bound C - W-bound C;
  set PW = p`1 - W;
  set KI = [\ (PW / EW * 2|^n) /];
  let I be Integer;
A1: EW > 0 by TOPREAL5:17,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`1;
  hence thesis by A3,A2,A4,XREAL_1:6;
end;
