
theorem Th26:
  for n being non zero Element of NAT holds n-roots_of_1 = {[**
  cos((2*PI*k)/n),sin((2*PI*k)/n)**] where k is Element of NAT: k < n }
proof
  let n be non zero Element of NAT;
  set X={[**cos((2*PI*k)/n),sin((2*PI*k)/n)**] where k is Element of NAT: k <
  n };
  now
    let x be object;
    hereby
      assume
A1:   x in n-roots_of_1;
      then reconsider a=x as Element of F_Complex;
      consider k being Element of NAT such that
A2:   a = [** cos((2*PI*k)/n), sin((2*PI*k)/n) **] by A1,Th24;
A3:   k mod n < n by NAT_D:1;
      a = [** cos((2*PI*(k mod n))/n), sin((2*PI*(k mod n))/n) **] by A2,Th10;
      hence x in X by A3;
    end;
    assume x in X;
    then
    ex k being Element of NAT st x = [**cos((2*PI*k)/n),sin((2*PI*k)/n)**]
    & k < n;
    hence x in n-roots_of_1 by Th24;
  end;
  hence thesis by TARSKI:2;
end;
