 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 Th19:
  for a being Element of A
  for g being Element of G
  holds (phi.(a")) . ((phi.a) . g) = g
  & (phi.a) . ((phi.(a")) . g) = g
proof
  let a be Element of A;
  let g be Element of G;
  thus (phi . (a")) . ((phi . a) . g) = (phi . (a" * a)) . g by Th18
                                     .= (phi . 1_A) . g by GROUP_1:def 5
                                     .= g by Th15;
  thus (phi. a) . ((phi . (a")) . g) = (phi . (a * a")) . g by Th18
                                    .= (phi . 1_A) . g by GROUP_1:def 5
                                    .= g by Th15;
end;
