
theorem Th28:
  for n, ni being non zero Element of NAT st ni divides n holds
  ni-roots_of_1 c= n-roots_of_1
proof
  let n,ni be non zero Element of NAT;
  assume ni divides n;
  then consider k being Nat such that
A1: n = ni*k by NAT_D:def 3;
  reconsider k as Element of NAT by ORDINAL1:def 12;
  for x being object st x in ni-roots_of_1 holds x in n-roots_of_1
  proof
    let x be object such that
A2: x in ni-roots_of_1;
    reconsider y=x as Element of F_Complex by A2;
    y is CRoot of ni,1_F_Complex by A2,Th21;
    then 1_F_Complex = (power F_Complex).(y, ni) by COMPLFLD:def 2;
    then 1_F_Complex=(power F_Complex).((power F_Complex).(y,ni),k) by Th8;
    then 1_F_Complex = (power F_Complex).(y,n) by A1,Th12;
    then y is CRoot of n,1_F_Complex by COMPLFLD:def 2;
    hence thesis;
  end;
  hence thesis;
end;
