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 Th49:
  for G being Group
  for N being strict normal Subgroup of G
  for phi being Automorphism of G
  holds Image(phi|N) is normal Subgroup of G
proof
  let G be Group;
  let N be strict normal Subgroup of G;
  let phi be Automorphism of G;
  set H = Image(phi|N);
  for g being Element of G holds g * H = H * g
  proof
    let g be Element of G;
    set f = (phi").g;
    B1: phi.f = g by Th4;
    B2: phi .: (f * N) = (phi.f) * (phi .: N) by Th47
                      .= g * H by B1,GRSOLV_1:def 3;
    phi .: (N * f) = (phi .: N)*(phi.f) by Th48
                  .= (phi .: N)*g by Th4
                  .= H*g by GRSOLV_1:def 3;
    hence g * H = H * g by B2,GROUP_3:117;
  end;
  hence H is normal Subgroup of G by GROUP_3:117;
end;
