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

theorem ThCapLemma: :: TH4
  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 Z1: the multMagma of H1 = the multMagma of K1;
  assume Z2: the multMagma of H2 = the multMagma of K2;
  Z3: K1 /\ K2 is Subgroup of G by GROUP_2:56;
  Z4: for g being Element of G
  st g in H1 /\ H2 holds g in K1 /\ K2
  proof
    let g be Element of G;
    assume g in H1 /\ H2;
    then g in H1 & g in H2 by GROUP_2:82;
    then g in K1 & g in K2 by Z1,Z2;
    hence g in K1 /\ K2 by GROUP_2:82;
  end;

  for g being Element of G
  st g in K1 /\ K2 holds g in H1 /\ H2
  proof
    let g be Element of G;
    assume g in K1 /\ K2;
    then g in K1 & g in K2 by GROUP_2:82;
    then g in H1 & g in H2 by Z1,Z2;
    hence g in H1 /\ H2 by GROUP_2:82;
  end;
  hence H1 /\ H2 = K1 /\ K2 by Z3, Z4, GROUP_2:def 6;
end;
