 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 Th27:
  for a being Element of A
  for x being Element of semidirect_product (G, A, phi)
  st x = <*1_G,a*>
  holds x" = <* 1_G, a " *>
proof
  let a be Element of A;
  let x be Element of semidirect_product (G, A, phi);
  assume A1: x = <*1_G,a*>;
  reconsider phi1=phi.(a ") as Homomorphism of G,G by AUTGROUP:def 1;
  thus x" = <* (phi.(a ")).((1_G) "), a " *> by A1, Th22
         .= <* phi1.(1_ G), a" *> by GROUP_1:8
         .= <* 1_G, a" *> by GROUP_6:31;
end;
