
theorem Th16:
  for n being Nat holds for x1,x2 being Element of
dyadic(n+1) st x1 < x2 & not x1 in dyadic(n) & not x2 in dyadic(n) holds
(axis(x1)+1)/(2|^(n+1)) <= (axis(x2)-1)/(2|^(n+1))
proof
  let n be Nat;
  let x1,x2 be Element of dyadic(n+1);
  assume that
A1: x1 < x2 and
A2: not x1 in dyadic(n) and
A3: not x2 in dyadic(n);
  consider k2 being Element of NAT such that
A4: axis(x2) = 2 * k2 or axis(x2) = 2 * k2 + 1 by SCHEME1:1;
A5: axis(x2) <> k2 * 2
  proof
    assume
A6: axis(x2) = k2 * 2;
    then x2 = (k2 * 2)/(2|^(n+1)) by Th10;
    then
A7: x2 = (k2 * 2)/((2|^n) * 2) by NEWTON:6
      .= (k2/(2|^n))*(2/2) by XCMPLX_1:76
      .= k2/(2|^n);
    k2 * 2 <= (2|^(n+1)) by A6,Th10;
    then k2 * 2 <= (2|^n) * 2 by NEWTON:6;
    then k2 <= ((2|^n)* 2)/2 by XREAL_1:77;
    hence thesis by A3,A7,Def1;
  end;
  consider k1 being Element of NAT such that
A8: axis(x1) = 2 * k1 or axis(x1) = 2 * k1 + 1 by SCHEME1:1;
A9: not axis(x1) = k1 * 2
  proof
    assume
A10: axis(x1) = k1 * 2;
    then x1 = (k1 * 2)/(2|^(n+1)) by Th10;
    then
A11: x1 = (k1 * 2)/((2|^n) * 2) by NEWTON:6
      .= (k1/(2|^n))*(2/2) by XCMPLX_1:76
      .= k1/(2|^n);
    k1 * 2 <= (2|^(n+1)) by A10,Th10;
    then k1 * 2 <= (2|^n) * 2 by NEWTON:6;
    then k1 <= ((2|^n)* 2)/2 by XREAL_1:77;
    hence thesis by A2,A11,Def1;
  end;
  then k1 * 2 + 1 < k2 * 2 + 1 by A1,A8,A4,A5,Th14;
  then k1 * 2 + 1 + (-1) < k2 * 2 + 1 + (-1) by XREAL_1:6;
  then (k1 * 2)/2 < (k2 * 2)/2 by XREAL_1:74;
  then k1 + 1 <= k2 by NAT_1:13;
  then 0 < (2|^(n+1)) & (k1 + 1) * 2 <= k2 * 2 by NEWTON:83,XREAL_1:64;
  hence thesis by A8,A4,A9,A5,XREAL_1:72;
end;
