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;

theorem Th23:
  for G being non trivial Group
  for H being Subgroup of G
  for phi being Automorphism of G
  st H is proper Subgroup of G
  holds Image(phi|H) is proper Subgroup of G
proof
  let G be non trivial Group;
  let H be Subgroup of G;
  let phi be Automorphism of G;
  set UH = the carrier of H;
  set UG = the carrier of G;
  A1: phi is one-to-one & phi is onto & UH is non empty Subset of UG &
      phi is Function of UG,UG by GROUP_2:def 5;
  assume H is proper Subgroup of G;
  then UG \ UH is non empty by Th11;
  then consider x such that
  A2: x in UG \ UH by XBOOLE_0:def 1;
  A3: x in G & not x in H by A2,XBOOLE_0:def 5;
  A4: not (phi.x in phi .: H)
      proof
        not (phi.x in (phi .: UH)) by A1, A3, Th5;
        hence not (phi.x in (phi .: H)) by GRSOLV_1:8;
      end;
  phi.x is Element of G
  proof
    dom phi = UG & rng phi = UG by A1, FUNCT_2:def 1;
    hence phi.x is Element of G by A2, FUNCT_1:def 3;
  end;
  then phi .: H is proper by A4;
  hence Image(phi|H) is proper Subgroup of G by GRSOLV_1:def 3;
end;
