
theorem Th32:
  for n being non zero Element of NAT holds n-roots_of_1 c= the
  carrier of MultGroup F_Complex
proof
  let n be non zero Element of NAT;
  set cMGFC = the carrier of MultGroup F_Complex;
  set FC = F_Complex;
  let a be object;
  assume a in n-roots_of_1;
  then consider x being Element of F_Complex such that
A1: a = x and
A2: x is CRoot of n,1_F_Complex;
  (power FC).(x,n) = 1_FC by A2,COMPLFLD:def 2;
  then x <> 0.FC by VECTSP_1:36;
  then
A3: not x in {0.FC} by TARSKI:def 1;
  cMGFC = NonZero FC by Def1;
  hence thesis by A1,A3,XBOOLE_0:def 5;
end;
