
theorem Th14:
  for G being cyclic finite Group, n,m being Nat st card G = n*m
  ex a being Element of G st ord a = n & gr {a} is strict Subgroup of G
proof
  let G be cyclic finite Group;
  let n,m be Nat;
  consider g be Element of G such that
A1: ord g = card G by GR_CY_1:19;
A2: m in NAT by ORDINAL1:def 12;
A3: n in NAT by ORDINAL1:def 12;
  assume card G = n*m;
  then ord (g|^m) = n by A1,A2,A3,INT_7:30;
  then consider a be Element of G such that
A4: ord a = n;
  take a;
  thus thesis by A4;
end;
