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;
reserve K for characteristic Subgroup of G;

theorem Th44:
  for G being Group
  for phi being Automorphism of G
  for H being Subgroup of G
  st (for h being Element of H holds phi.h in H)
  holds Image(phi|H) is Subgroup of H
proof
  let G be Group;
  let phi be Automorphism of G;
  let H be Subgroup of G;
  assume A1: for h being Element of H holds phi.h in H;
  for y being object st y in rng(phi|H) holds y in the carrier of H
  proof
    let y be object;
    assume y in rng(phi|H);
    then consider x being object such that
    B1: x in dom(phi|H) and
    B2: y = (phi|H).x
    by FUNCT_1:def 3;
    B3: x in H by B1;
    reconsider x as Element of H by B1;
    phi.x in H & x is Element of G by A1,GROUP_2:42;
    hence y in the carrier of H by B2,B3,Th1;
  end;
  then rng(phi|H) c= the carrier of H;
  then the carrier of Image(phi|H) c= the carrier of H by GROUP_6:44;
  hence Image(phi|H) is Subgroup of H by GROUP_2:57;
end;
