
theorem Th34:
  for n being non zero Element of NAT, x being Element of
  MultGroup F_Complex holds ord x divides n iff x in n-roots_of_1
proof
  set FC = F_Complex;
  let n be non zero Element of NAT, x be Element of MultGroup F_Complex;
  reconsider c = x as Element of FC by Th19;
  set MGFC = MultGroup F_Complex;
A1: 1_MGFC = 1_FC by Th17;
  hereby
    assume ord x divides n;
    then consider k being Nat such that
A2: n = (ord x)*k by NAT_D:def 3;
    x |^ ord x = 1_MGFC by GROUP_1:41;
    then (x |^ (ord x)) |^ k = 1_MGFC by GROUP_1:31;
    then x |^ n = 1_MGFC by A2,GROUP_1:35;
    then (power FC).(c, n) = 1_FC by A1,Th29;
    then c is CRoot of n,1_FC by COMPLFLD:def 2;
    hence x in n-roots_of_1;
  end;
  assume x in n-roots_of_1;
  then consider c being Element of FC such that
A3: c = x and
A4: c is CRoot of n,1_FC;
  (power FC).(c,n) = 1_FC by A4,COMPLFLD:def 2;
  then x |^ n = 1_MGFC by A1,A3,Th29;
  hence thesis by GROUP_1:44;
end;
