 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 Th33:
  (Image incl1(G,A,phi)) * (Image incl2(G,A,phi))
  = the carrier of semidirect_product(G,A,phi)
proof
  for x being Element of semidirect_product (G, A, phi)
  holds x in (Image incl1(G,A,phi)) * (Image incl2(G,A,phi))
  proof
    let x be Element of semidirect_product (G, A, phi);

    consider g being Element of G, a being Element of A such that
    B1: ((incl1(G,A,phi)).g) * ((incl2(G,A,phi)).a) = x
      by Th32;
    ((incl1(G,A,phi)).g) in Image(incl1(G,A,phi)) &
    ((incl2(G,A,phi)).a) in Image(incl2(G,A,phi)) by GROUP_6:45;
    hence x in (Image incl1(G,A,phi)) * (Image incl2(G,A,phi))
      by B1, GROUP_5:4;
  end;
  then the carrier of (semidirect_product(G,A,phi))
    c= (Image incl1(G,A,phi)) * (Image incl2(G,A,phi));
  hence thesis by XBOOLE_0:def 10;
end;
