theorem
  for Gc being strict finite cyclic Group st ex k st card Gc = 2*k holds
  ex H being Subgroup of Gc st card H = 2 & H is cyclic Group
proof
  let Gc be strict finite cyclic Group;
  set n = card Gc;
  assume ex k st n=2*k;
  then consider g1 being Element of Gc such that
A1: ord g1 = 2 and
  for g2 being Element of Gc st ord g2=2 holds g1=g2 by Th24;
  take gr {g1};
  thus thesis by A1,Th4,GR_CY_1:7;
end;
