
theorem
  for n,q being non zero Element of NAT, qc being Element of F_Complex
  st qc = q for j being Integer st j = eval(cyclotomic_poly(n),qc) holds j
  divides (q |^ n - 1)
proof
  let n,q be non zero Element of NAT;
  let qc be Element of F_Complex such that
A1: qc = q;
A2: ex y1 being Element of F_Complex st y1 = q & eval( unital_poly(F_Complex,
  n),y1) = (q |^ n) - 1 by Th44;
  let j be Integer;
  assume j = eval(cyclotomic_poly(n),qc);
  hence thesis by A1,A2,Th56;
end;
