
theorem Th42:
  for n being non zero Element of NAT holds Roots unital_poly(
  F_Complex, n) = n-roots_of_1
proof
  let n be non zero Element of NAT;
  now
    set p = unital_poly(F_Complex,n);
    let x be object;
    hereby
      assume
A1:   x in Roots p;
      then reconsider x9 = x as Element of F_Complex;
      x9 is_a_root_of p by A1,POLYNOM5:def 10;
      then 0.F_Complex = eval(p,x9) by POLYNOM5:def 7
        .= (power F_Complex).(x9,n) - 1 by Th41;
      then x9 is CRoot of n, 1_F_Complex by COMPLFLD:7,8,def 2;
      hence x in n-roots_of_1;
    end;
    assume
A2: x in n-roots_of_1;
    then reconsider x9 = x as Element of F_Complex;
    x9 is CRoot of n, 1_F_Complex by A2,Th21;
    then (power F_Complex).(x9,n) = 1 by COMPLFLD:8,def 2;
    then (power F_Complex).(x9,n) - 1 = 0c;
    then eval(p,x9) = 0.F_Complex by Th41,COMPLFLD:7;
    then x9 is_a_root_of p by POLYNOM5:def 7;
    hence x in Roots unital_poly(F_Complex,n) by POLYNOM5:def 10;
  end;
  hence thesis by TARSKI:2;
end;
