 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 Th22:
  for a being Element of A
  for g being Element of G
  for x being Element of semidirect_product (G, A, phi)
  st x = <* g, a *>
  holds x" = <* (phi.(a")).(g"), a" *>
proof
  set S = semidirect_product (G, A, phi);
  let a be Element of A;
  let g be Element of G;
  let x be Element of semidirect_product (G, A, phi);
  assume A1: x = <* g, a *>;
  reconsider phi1=phi.(a ") as Homomorphism of G,G by AUTGROUP:def 1;
  reconsider y = <* phi1.(g"), a" *> as Element of S by Th9;
  x * y = 1_S & y * x = 1_S by A1, Th20;
  hence thesis by GROUP_1:5;
end;
