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 Th16:
  (for f being Automorphism of G holds Image(f|H) is Subgroup of H) implies
  ex psi being Automorphism of G
  st psi = phi" & Image(phi|Image(psi|H)) is Subgroup of Image(phi|H)
proof
  assume A1: for f being Automorphism of G holds Image(f|H) is Subgroup of H;
  reconsider psi = phi" as Automorphism of G by GROUP_6:62;
  take psi;
  thus psi = phi";
  Image(psi|H) is Subgroup of H by A1;
  then phi .: Image(psi|H) is Subgroup of phi .: H by GRSOLV_1:12;
  then Image(phi|Image(psi|H)) is Subgroup of phi .: H by GRSOLV_1:def 3;
  hence Image(phi|Image(psi|H)) is Subgroup of Image(phi|H) by GRSOLV_1:def 3;
end;
