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 Th24:
  for G being non trivial Group
  for H being strict Subgroup of G
  for phi being Automorphism of G
  st H is maximal
  holds Image(phi|H) is maximal
proof
  let G be non trivial Group;
  let H be strict Subgroup of G;
  let phi be Automorphism of G;
  assume A1: H is maximal;
  A2: Image(phi|H) is proper Subgroup of G by A1,Th23;
  set UG = the carrier of G;
  set UH = the carrier of H;
  for K being strict Subgroup of G
  st Image(phi|H) <> K & Image(phi|H) is Subgroup of K
  holds K = the multMagma of G
  proof
    let K be strict Subgroup of G;
    assume B1: Image(phi|H) <> K;
    assume B2: Image(phi|H) is Subgroup of K;
    then consider psi being Automorphism of G such that
    B3: psi = phi" and
    B4: H is Subgroup of Image(psi|K)
    by Th18;
    set UK = the carrier of K;
    reconsider K as non trivial strict Subgroup of G by A2,B1,B2,Th12;
    UK \ (the carrier of Image(phi|H)) is non empty by B1,B2,Def1,Th11;
    then consider k being object such that
    B6: k in UK \ (the carrier of Image(phi|H)) by XBOOLE_0:def 1;
    reconsider k as Element of K by B6;
    set L = Image(psi|K);
    B8: psi.k in L
    proof
      C1: k in G by GROUP_2:41;
      consider l being object such that
      C2: l = psi.k;
      dom psi = the carrier of G by FUNCT_2:def 1;
      then l in psi .: (the carrier of K) by C1,C2,FUNCT_1:def 6;
      then l in the carrier of (psi .: K) by GRSOLV_1:8;
      hence psi.k in Image(psi|K) by C2,GRSOLV_1:def 3;
    end;
    B9: the multMagma of H <> L
    proof
      set UPH = the carrier of Image(phi|H);
      C1: phi is one-to-one & phi is onto & UPH is non empty Subset of UG &
      phi is Function of UG,UG by GROUP_2:def 5;
      C2: k in G & not k in Image(phi|H) by B6, XBOOLE_0:def 5, GROUP_2:41;
      consider phi2 being Automorphism of G such that
      C3: phi2 = psi" and
      C4: Image(psi|Image(phi2|H)) = the multMagma of H
      by Th17;
      C5: phi2=phi by C3,B3,FUNCT_1:43;
      set UPH = the carrier of Image(phi|H);
      psi .: UPH = the carrier of (psi .: Image(phi|H)) by GRSOLV_1:8
                .= the carrier of Image(psi|Image(phi|H)) by GRSOLV_1:def 3;
      hence thesis by B8,C1,C2,C4,C5,Th5;
    end;
    B10: Image(phi|L) = the multMagma of G
    proof
      L = the multMagma of G by A1,B4,B9,GROUP_4:def 6;
      then phi .: the carrier of L = rng phi by RELSET_1:22
                                  .= UG by FUNCT_2:def 3;
      then UG = the carrier of (phi .: L) by GRSOLV_1:8
             .= the carrier of Image(phi|L) by GRSOLV_1:def 3;
      hence thesis by GROUP_2:61;
    end;
    Image(phi|L) = K
    proof
      consider psi2 being Automorphism of G such that
      C1: psi2 = phi" and
      C2: Image(phi|Image(psi2|K)) = the multMagma of K by Th17;
      thus Image(phi|Image(psi|K)) = K by B3,C1,C2;
    end;
    hence thesis by B10;
  end;
  hence Image(phi|H) is maximal by A2,Def1,GROUP_4:def 6;
end;
