
theorem Th35:
  for n being non zero Element of NAT holds n-roots_of_1 = { x
  where x is Element of MultGroup F_Complex : ord x divides n}
proof
  set cMGFC = the carrier of MultGroup F_Complex;
  set MGFC = MultGroup F_Complex;
  let n be non zero Element of NAT;
  set R = { a where a is Element of F_Complex : a is CRoot of n,1_F_Complex };
  set S = {x where x is Element of MultGroup F_Complex : ord x divides n};
A1: n-roots_of_1 = R;
  then
A2: R c= cMGFC by Th32;
  now
    let a be object;
    hereby
      assume
A3:   a in R;
      then reconsider x = a as Element of MGFC by A2;
      ord x divides n by A1,A3,Th34;
      hence a in S;
    end;
    assume a in S;
    then ex x being Element of MGFC st a = x & ord x divides n;
    hence a in R by A1,Th34;
  end;
  hence thesis by TARSKI:2;
end;
