 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);
 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);

theorem Th30:
  Image (incl1 (G, A, phi)) is normal Subgroup of
  semidirect_product (G, A, phi)
proof
  for x,g being Element of semidirect_product (G, A, phi)
  st g is Element of Image incl1 (G, A, phi)
  holds g |^ x in Image incl1 (G, A, phi)
  proof
    let x,g be Element of semidirect_product (G, A, phi);
    assume g is Element of Image incl1 (G, A, phi); then
    g in Image incl1 (G, A, phi);
    then consider g0 being Element of G such that
    A1: g = (incl1 (G, A, phi)).g0 by GROUP_6:45;
    A2: g = <*g0,1_A*> by A1,Def2;
    consider x1 being Element of G, x2 being Element of A such that
    A3: x = <* x1, x2 *> by Th12;
    reconsider phi2=phi.(x2 ") as Homomorphism of G,G by AUTGROUP:def 1;
    x" = <* (phi2).(x1"),x2" *> by A3, Th22;
    then (x") * g = <* ((phi2).(x1"))*((phi2).g0),
                       (x2") * (1_A) *> by A2,Th14
    .= <* ((phi2).(x1"))*((phi2).g0), (x2") *> by GROUP_1:def 4
    .= <* (phi2).((x1") * g0), (x2") *> by GROUP_6:def 6;
    then A4: ((x") * g) * x
    = <* ((phi2).((x1")*g0)*((phi2).x1)), (x2") * x2 *> by A3,Th14
    .= <* ((phi2).((x1")*g0)*((phi2).x1)), 1_A *>
       by GROUP_1:def 5
    .= <* ((phi2).(((x1") * g0) * x1)), 1_A *> by GROUP_6:def 6
    .= <* (phi2).(g0 |^ x1), 1_A *> by GROUP_3:def 2;
    g |^ x = ((x") * g) * x by GROUP_3:def 2
    .= ((incl1 (G, A, phi)).((phi2).(g0 |^ x1))) by A4,Def2;
    hence g |^ x in Image (incl1 (G, A, phi)) by GROUP_6:45;
  end;
  hence thesis by AUTGROUP:1;
end;
