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

theorem ThProdLemma: :: TH5
  for G being Group
  for H1,H2,K being Subgroup of G
  for K1,K2 being Subgroup of K
  st the multMagma of H1 = the multMagma of K1
  & the multMagma of H2 = the multMagma of K2
  holds H1 * H2 = K1 * K2
proof
  let G be Group;
  let H1,H2,K be Subgroup of G;
  let K1,K2 be Subgroup of K;
  assume A1: the multMagma of H1 = the multMagma of K1;
  assume A2: the multMagma of H2 = the multMagma of K2;
  A3: carr H1 = carr K1 by A1;
  A4: carr H2 = carr K2 by A2;
  for x being object
  holds x in (carr H1) * (carr H2) iff x in (carr K1) * (carr K2)
  proof
    let x be object;
    thus x in (carr H1) * (carr H2) implies x in (carr K1) * (carr K2)
    proof
      assume x in (carr H1) * (carr H2);
      then consider h1,h2 being Element of G such that
      B1: x = h1*h2 & h1 in carr H1 & h2 in carr H2;
      reconsider k1=h1,k2=h2 as Element of K by A3,A4,B1;
      k1*k2 = x by B1,GROUP_2:43;
      hence x in (carr K1) * (carr K2) by A3, A4, B1;
    end;

    assume x in (carr K1) * (carr K2);
    then consider k1,k2 being Element of K such that
    B1: x = k1*k2 & k1 in carr K1 & k2 in carr K2;
    reconsider h1=k1,h2=k2 as Element of G by GROUP_2:42;
    h1 * h2 = x by B1,GROUP_2:43;
    hence x in (carr H1) * (carr H2) by A3, A4, B1;
  end;
  then A5: (carr H1) * (carr H2) = (carr K1) * (carr K2) by TARSKI:2;

  thus H1 * H2 = (carr H1) * (carr H2) by GROUP_4:def 8
              .= K1 * K2 by A5,GROUP_4:def 8;
end;
