 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 Th36:
  for a being Element of A
  for g being Element of G
  holds ((incl1(G,A,phi)).g) |^ ((incl2(G,A,phi)).a)
  = <*(phi.(a")).g,1_A *>
proof
  let a be Element of A;
  let g be Element of G;
  set xg = (incl1(G,A,phi)).g;
  set xa = (incl2(G,A,phi)).a;
  reconsider phi1=phi.(a") as Homomorphism of G,G by AUTGROUP:def 1;
  A1: xa = <* 1_G,a *> by Def3;
  then A2: xa" = <* (phi1.((1_G)")), a" *> by Th22
              .= <* (phi1.(1_G)), a" *> by GROUP_1:8
              .= <* 1_G, a" *> by GROUP_6:31;
  xg = <*g, 1_A*> by Def2; then
  (xa") * xg = <* (1_G) * (phi1.g), (a") * (1_A) *> by A2,Th14
            .= <* (1_G) * (phi1.g), (a") *> by GROUP_1:def 4
            .= <* (phi1.g), a" *> by GROUP_1:def 4;
  then ((xa")*xg)*xa = <* (phi1.g)*(phi1.(1_G)), (a")*a *>
  by A1, Th14
                    .= <* (phi1.g) * (1_G), (a")*a *> by GROUP_6:31
                    .= <* (phi1.g), (a")*a *> by GROUP_1:def 4
                    .= <* (phi1.g), 1_A *> by GROUP_1:def 5;
  hence thesis by GROUP_3:def 2;
end;
