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

theorem Th7:
  for G1,G2 being Group
  for H1 being Subgroup of G1
  for H2 being Subgroup of G2
  for h1 being Element of G1 st h1 in H1
  for h2 being Element of G2 st h2 in H2
  holds <*h1,h2*> in product <*H1,H2*>
proof
  let G1,G2 be Group;
  let H1 be Subgroup of G1;
  let H2 be Subgroup of G2;
  let h1 be Element of G1;
  assume A1: h1 in H1;
  let h2 be Element of G2;
  assume A2: h2 in H2;
  set H = <*H1,H2*>;
  A3: dom Carrier H = {1,2} by PARTFUN1:def 2;
  A4: for a being object st a in {1,2}
      holds <*h1,h2*>.a in (Carrier <*H1,H2*>).a
  proof
    let a be object;
    assume B1: a in {1,2};
    then per cases by TARSKI:def 2;
    suppose B2: a = 1;
      then ex R being 1-sorted st R = H.1 &
        (Carrier H).1 = the carrier of R
      by B1, PRALG_1:def 15;
      hence thesis by A1,B2;
    end;
    suppose B4: a = 2;
      then ex R being 1-sorted st R = H.2 &
        (Carrier H).2 = the carrier of R
      by B1, PRALG_1:def 15;
      hence thesis by A2,B4;
    end;
  end;
  dom <*h1,h2*> = {1,2} by FINSEQ_1:2,89;
  then <*h1,h2*> in product Carrier <*H1,H2*> by A3,A4,CARD_3:9;
  hence <*h1,h2*> in product <*H1,H2*> by GROUP_7:def 2;
end;
