 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 Th23:
  for g1,g2 being Element of G
  for x,y,z being Element of semidirect_product (G, A, phi)
  st x = <*g1,1_A*> & y = <*g2,1_A*> & z = <*(g1*g2),1_A*>
  holds x * y = z
proof
  let g1,g2 be Element of G;
  let x,y,z be Element of semidirect_product (G, A, phi);
  assume A1: x = <*g1,1_A*>;
  assume A2: y = <*g2,1_A*>;
  assume A3: z = <*(g1*g2),1_A*>;
  reconsider phi1 = phi.(1_A) as Homomorphism of G,G by AUTGROUP:def 1;
  thus x * y = <* g1 * (phi1 . g2), (1_A) * (1_A) *> by A1,A2,Th14
            .= <* g1 * (phi1 . g2), 1_A *> by GROUP_1:def 4
            .= z by A3, Th15;
end;
