 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 :: TH34
  Image (incl1 (G, A, phi)) "\/" Image (incl2 (G, A, phi))
  = (semidirect_product(G,A,phi))
proof
  set S = semidirect_product(G,A,phi);
  set U = (Image incl1(G,A,phi)) * (Image incl2(G,A,phi));
  U c= the carrier of gr U by GROUP_4:def 4; then
  the carrier of S c= the carrier of (gr U) by Th33;
  then S = gr U by GROUP_2:61;
  hence thesis by GROUP_4:50;
end;
