
theorem Th52: :: WEDDWITT
  for d being non zero Element of NAT, z being Element of
  F_Complex st z is Integer holds eval(cyclotomic_poly(d),z) is Integer
proof
  let d be non zero Element of NAT,z be Element of F_Complex such that
A1: z is Integer;
  set phi = cyclotomic_poly(d);
  consider F being FinSequence of F_Complex such that
A2: eval(phi,z) = Sum F and
  len F = len phi and
A3: for i being Element of NAT st i in dom F holds F.i = phi.(i-'1) * (
  power F_Complex).(z,i-'1) by POLYNOM4:def 2;
  for i being Element of NAT st i in dom F holds F.i is Integer
  proof
    let i be Element of NAT;
    assume i in dom F;
    then
A4: F.i = phi.(i-'1) * (power F_Complex).(z,i-'1) by A3;
    reconsider i2 = (power F_Complex).(z,i-'1) as Integer by A1,Th13;
    reconsider i1 = phi.(i-'1) as Integer by Th51;
    F.i = i1*i2 by A4;
    hence thesis;
  end;
  hence thesis by A2,Th14;
end;
