
theorem
  for n,ni,q being non zero Element of NAT st ni < n & ni divides n for
  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) div (q |^ ni - 1))
proof
  let n,ni,q be non zero Element of NAT such that
A1: ni < n & ni divides n;
  let qc be Element of F_Complex such that
A2: qc = q;
A3: (ex y1 being Element of F_Complex st y1 = q & eval( unital_poly(
F_Complex,n) ,y1) = (q |^ n) - 1 )& ex y2 being Element of F_Complex st y2=q &
  eval( unital_poly(F_Complex,ni),y2) = (q |^ ni) - 1 by Th44;
  let j be Integer;
  assume j = eval(cyclotomic_poly(n),qc);
  hence thesis by A1,A2,A3,Th57;
end;
