
theorem
  for n being non zero Element of NAT holds n-th_roots_of_1 is Subgroup
  of MultGroup F_Complex
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;
  set nth = n-th_roots_of_1;
  set cnth = the carrier of nth;
A1: the carrier of nth = n-roots_of_1 by Def3;
  then
A2: the carrier of nth c= the carrier of MGFC by Th32;
  the multF of nth = (the multF of FC)||(n-roots_of_1) & the multF of MGFC
  = ( the multF of FC)||cMGFC by Def1,Def3;
  then the multF of nth = (the multF of MGFC)||cnth by A1,A2,RELAT_1:74
,ZFMISC_1:96;
  hence thesis by A2,GROUP_2:def 5;
end;
