 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 Th18:
  for a1,a2 being Element of A
  for g being Element of G
  holds (phi.a1) . ((phi . a2) . g) = (phi . (a1 * a2)) . g
proof
  let a1,a2 be Element of A;
  let g be Element of G;

  reconsider phi1 = phi.a1,phi2=phi.a2 as Homomorphism of G,G
    by AUTGROUP:def 1;
  phi . (a1 * a2) = (phi.a1) * (phi.a2) by GROUP_6:def 6;
  then (phi . (a1 * a2)) . g = ((phi1) * (phi2)).g by AUTGROUP:8
                            .= (phi1) . ((phi2) . g) by FUNCT_2:15;
  hence (phi.a1) . ((phi . a2) . g) = (phi . (a1 * a2)) . g;
end;
