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 Th22:
  for f being Automorphism of G
  st Image(f|H) = the multMagma of H
  holds f|H is Automorphism of H
proof
  let f be Automorphism of G;
  assume A1: Image(f|H) = the multMagma of H;
  set UH = the carrier of H;
  reconsider fH=f|H as Function of UH,UH by A1,GROUP_6:49;
  A2: fH is bijective
  proof
    thus fH is one-to-one by Lm3;
    UH = rng(f|H) by A1, GROUP_6:44
      .= rng(fH);
    hence fH is onto;
  end;
  for x,y being Element of H holds fH.(x*y) = (fH.x) * (fH.y)
  proof
    let x,y be Element of H;
    fH.(x * y) = (f|H).x * (f|H).y by GROUP_6:def 6
              .= fH.x * fH.y by GROUP_2:43;
    hence thesis;
  end;
  hence thesis by A2,GROUP_6:def 6;
end;
