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

theorem Th12:
  for C being compact non horizontal Subset of TOP-REAL 2 for J
being Integer st p in BDD C & J = [\ ((p`2 - S-bound C) / (N-bound C - S-bound
  C) * 2|^n) + 2 /] holds 1 < J & J+1 <= width Gauge (C, n)
proof
  let C be compact non horizontal Subset of TOP-REAL 2;
  set W = S-bound C, E = N-bound C;
  set EW = E-W, pW = p`2 - W;
  let I be Integer;
  assume that
A1: p in BDD C and
A2: I = [\ (pW / EW * 2|^n) + 2 /];
A3: EW > 0 by SPRECT_1:32,XREAL_1:50;
  set K = [\ pW / EW * 2|^n /];
  pW / EW * 2|^n - 1 < K by INT_1:def 6;
  then
A4: pW / EW * 2|^n - 1 + 2 < K + 2 by XREAL_1:6;
A5: W <= S-bound BDD C by A1,Th8;
  BDD C is bounded by JORDAN2C:106;
  then p`2 >= S-bound BDD C by A1,Th5;
  then p`2 >= W by A5,XXREAL_0:2;
  then pW >= 0 by XREAL_1:48;
  then pW / EW * 2|^n + 1 >= 0 + 1 by A3,XREAL_1:6;
  then 1 < K + 2 by A4,XXREAL_0:2;
  hence 1 < I by A2,INT_1:28;
A6: len Gauge (C, n) = width Gauge (C, n) by JORDAN8:def 1;
A7: E >= N-bound BDD C by A1,Th9;
  BDD C is bounded by JORDAN2C:106;
  then p`2 <= N-bound BDD C by A1,Th5;
  then p`2 <= E by A7,XXREAL_0:2;
  then p`2 - W <= EW by XREAL_1:9;
  then pW / EW <= 1 by A3,XREAL_1:185;
  then pW / EW * 2|^n <= 1 * 2|^n by XREAL_1:64;
  then
A8: pW / EW * 2|^n + 3 <= 2|^n + 3 by XREAL_1:7;
  I <= (pW / EW * 2|^n) + 2 by A2,INT_1:def 6;
  then
A9: I + 1 <= (pW / EW * 2|^n) + 2 + 1 by XREAL_1:6;
  len Gauge (C, n) = 2|^n + 3 by JORDAN8:def 1;
  hence thesis by A6,A8,A9,XXREAL_0:2;
end;
