 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 Th12:
  for x being Element of semidirect_product (G, A, phi)
  ex g being Element of G, a being Element of A
  st x = <* g, a *>
proof
  let x be Element of semidirect_product (G, A, phi);

  x.1 in G & x.2 in A by Th11;
  then consider g being Element of G, a being Element of A such that
  A1: g = x.1 & a = x.2;

  take g, a;

  dom x = {1,2} by Th11;
  then len x = 2 by FINSEQ_1:2,def 3;
  hence x = <* g, a *> by A1, FINSEQ_1:44;
end;
