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 Th18:
  for H being strict Subgroup of G
  for K being Subgroup of G
  st Image(phi|H) is Subgroup of K
  holds ex psi being Automorphism of G
  st psi = phi" & H is Subgroup of Image(psi|K)
proof
  let H be strict Subgroup of G;
  let K be Subgroup of G;
  assume A1: Image(phi|H) is Subgroup of K;
  reconsider psi = phi" as Automorphism of G by GROUP_6:62;
  take psi;
  thus psi = phi";
  consider phi0 being Automorphism of G such that
  A2: phi0 = psi" and
  A3: Image(psi|Image(phi0|H)) = the multMagma of H by Th17;
  A4: phi = phi0 by A2,FUNCT_1:43;
  psi .: Image(phi|H) is Subgroup of psi .: K by A1,GRSOLV_1:12;
  then Image(psi|Image(phi|H)) is Subgroup of psi .: K by GRSOLV_1:def 3;
  hence H is Subgroup of Image(psi|K) by A3,A4,GRSOLV_1:def 3;
end;
