 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 Th24:
  for g being Element of G
  for x being Element of semidirect_product (G, A, phi)
  st x = <*g,1_A*>
  holds x" = <* g", 1_A *>
proof
  let g be Element of G;
  let x be Element of semidirect_product (G, A, phi);
  assume A1: x = <*g,1_A*>;
  reconsider phi1=phi.1_A,phi2=phi.((1_A) ") as Homomorphism of G,G
    by AUTGROUP:def 1;
  A2: phi.((1_A) ") = phi1 by GROUP_1:8;
  thus x" = <* (phi.((1_A) ")).(g "), (1_A) " *> by A1, Th22
         .= <* phi1.(g "), 1_A *> by A2,GROUP_1:8
         .= <* g", 1_A *> by Th15;
end;
