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

theorem Th10:
  for C being compact non vertical Subset of TOP-REAL 2 for I
being Integer st p in BDD C & I = [\ ((p`1 - W-bound C) / (E-bound C - W-bound
  C) * 2|^n) + 2 /] holds 1 < I
proof
  let C be compact non vertical Subset of TOP-REAL 2;
  set W = W-bound C, E = E-bound C;
  set pW = p`1 - W, EW = E - W;
  let I be Integer;
  assume that
A1: p in BDD C and
A2: I = [\ (pW / EW * 2|^n) + 2 /];
A3: W <= W-bound BDD C by A1,Th6;
  BDD C is bounded by JORDAN2C:106;
  then p`1 >= W-bound BDD C by A1,Th5;
  then p`1 >= W by A3,XXREAL_0:2;
  then
A4: pW >= 0 by XREAL_1:48;
  set K = [\ pW / EW * 2|^n /];
  pW / EW * 2|^n - 1 < K by INT_1:def 6;
  then
A5: pW / EW * 2|^n - 1 + 2 < K + 2 by XREAL_1:6;
  EW > 0 by SPRECT_1:31,XREAL_1:50;
  then pW / EW * 2|^n + 1 >= 0 + 1 by A4,XREAL_1:6;
  then 1 < K + 2 by A5,XXREAL_0:2;
  hence thesis by A2,INT_1:28;
end;
