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

theorem Th14:
  for I being Integer st I = [\ ((p`1 - W-bound C) / (E-bound C -
W-bound C) * 2|^n) + 2 /] holds p`1 < (W-bound C) + (((E-bound C)-(W-bound C))/
  (2|^n))*(I-1)
proof
  set W = W-bound C, E = E-bound C;
  set EW = E - W, PW = p`1 - 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: EW > 0 by TOPREAL5:17,XREAL_1:50;
  then EW/(2|^n) > 0 by A1,XREAL_1:139;
  then
A4: (EW/(2|^n))*KI > (EW/(2|^n))*(PW / EW * 2|^n) by A2,XREAL_1:68;
A5: W + PW = p`1;
  (EW/(2|^n))*(PW / EW * 2|^n) = PW by A3,A1,Lm2;
  hence thesis by A5,A4,XREAL_1:6;
end;
