theorem Th11:
  for x being Element of semidirect_product (G, A, phi)
  holds x.1 in G & x.2 in A & dom x = {1,2}
proof
  let x be Element of semidirect_product (G, A, phi);
  x in the carrier of semidirect_product (G, A, phi);
  then x in the carrier of product <*G,A*> by Th9;
  hence thesis by Th6;
end;
