 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 Th16:
  for x,y being Element of semidirect_product (G, A, phi)
  for a being Element of A
  for g being Element of G
  st x = <*g,1_A*> & y = <*1_G,a*>
  holds x * y = <* g, a *>
proof
  let x,y be Element of semidirect_product (G, A, phi);
  let a be Element of A;
  let g be Element of G;
  assume A1: x = <*g,1_A*>;
  assume A2: y = <*1_G,a*>;
  ((phi . (1_A)).(1_G)) = 1_G by Th15;
  hence x * y = <* g * (1_G), (1_A) * a *> by A1,A2,Th14
             .= <* g, (1_A)*a *> by GROUP_1:def 4
             .= <* g, a *> by GROUP_1:def 4;
end;
