reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem
  for G being non trivial Group
  holds (ex H being strict Subgroup of G st H is maximal) implies
  Phi(G) is characteristic Subgroup of G
proof
  let G be non trivial Group;
  defpred P[Subgroup of G] means $1 is maximal;
  assume A1: ex H being strict Subgroup of G st P[H];
  set MaxSubCarrs = {A where A is Subset of G :
                     ex H being strict Subgroup of G
                     st A = the carrier of H & P[H]};

  A2: for phi being Automorphism of G
  for H being strict Subgroup of G
  st P[H]
  holds P[Image(phi|H)] by Th24;

  consider K being strict Subgroup of G such that
  A3: the carrier of K = meet {A where A is Subset of G :
      ex H being strict Subgroup of G st A = the carrier of H & P[H]} and
  A4: K is characteristic
  from MeetIsChar(A2,A1);
  thus thesis by A4,A1,A3,GROUP_4:def 7;
end;
