
theorem Th30:
  for n being non zero Element of NAT, x being Element of
  MultGroup F_Complex st x in n-roots_of_1 holds x is not being_of_order_0
proof
  set MGFC = MultGroup F_Complex;
  set cMGFC = the carrier of MultGroup F_Complex;
  set FC = F_Complex;
  let n be non zero Element of NAT, x be Element of cMGFC;
  assume x in n-roots_of_1;
  then consider c being Element of FC such that
A1: c = x and
A2: c is CRoot of n,1_FC;
A3: 1_MGFC = 1_FC by Th17;
  (power FC).(c,n) = 1_FC by A2,COMPLFLD:def 2;
  then x |^ n = 1_MGFC by A1,A3,Th29;
  hence thesis;
end;
