 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 Th26:
  for a1,a2 being Element of A
  for x,y,z being Element of semidirect_product (G, A, phi)
  st x = <*1_G,a1*> & y = <*1_G,a2*> & z = <*1_G,(a1*a2)*>
  holds x * y = z
proof
  let a1,a2 be Element of A;
  let x,y,z be Element of semidirect_product (G, A, phi);
  assume A1: x = <*1_G, a1*>;
  assume A2: y = <*1_G, a2*>;
  assume A3: z = <*1_G, (a1*a2)*>;
  reconsider phi1 = phi.a1 as Homomorphism of G,G by AUTGROUP:def 1;
  thus x * y = <* 1_G * (phi1.(1_G)), a1 * a2 *> by A1,A2,Th14
            .= <* 1_G * (1_G), a1 * a2 *> by GROUP_6:31
            .= z by A3,GROUP_1:def 4;
end;
