 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 Th32:
  for x being Element of semidirect_product(G,A,phi)
  ex g being Element of G,a being Element of A
  st ((incl1(G,A,phi)).g) * ((incl2(G,A,phi)).a) = x
proof
  let x be Element of semidirect_product(G, A, phi);
  consider g being Element of G, a being Element of A such that
  A1: x = <*g,a*> by Th12;
  take g,a;
  reconsider phi1=phi.(1_A) as Homomorphism of G,G by AUTGROUP:def 1;
  ((incl1(G,A,phi)).g) = <*g,1_A*> & ((incl2(G,A,phi)).a) = <*1_G,a*>
    by Def2,Def3;
  hence ((incl1(G,A,phi)).g) * ((incl2(G,A,phi)).a)
  = <* g * (phi1.(1_G)), (1_A) * a *> by Th14
  .= <* g * (1_G), (1_A) * a *> by Th15
  .= <* g * (1_G), a *> by GROUP_1:def 4
  .= x by A1, GROUP_1:def 4;
end;
