 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);

theorem Th6:
  for G1,G2 being Group
  for x being Element of product <*G1,G2*>
  holds x.1 in G1 & x.2 in G2 & dom x = {1,2}
proof
  let G1,G2 be Group;
  let x be Element of product (<*G1,G2*>);
  A1: dom Carrier <*G1,G2*> = {1,2} by PARTFUN1:def 2;
  x in the carrier of product (<*G1,G2*>);
  then x in product Carrier <*G1,G2*> by GROUP_7:def 2;
  then consider f being Function such that
  A2: x = f
      & dom f = dom Carrier <*G1,G2*>
      & (for y being object st y in dom Carrier <*G1,G2*>
         holds f.y in (Carrier <*G1,G2*>).y)
  by CARD_3:def 5;

  thus x.1 in G1
  proof
    B2: 1 in dom Carrier <*G1,G2*> by A1, TARSKI:def 2;
    then ex R being 1-sorted st
    (R = <*G1,G2*>.1 & (Carrier <*G1,G2*>).1 = the carrier of R)
    by PRALG_1:def 15;
    hence thesis by A2, B2;
  end;

  thus x.2 in G2
  proof
    B2: 2 in dom (Carrier <*G1,G2*>) by A1, TARSKI:def 2;
    then ex R being 1-sorted st
    (R = <*G1,G2*>.2 & (Carrier <*G1,G2*>).2 = the carrier of R)
    by PRALG_1:def 15;
    hence thesis by A2, B2;
  end;
  thus dom x = {1,2} by A2,PARTFUN1:def 2;
end;
