reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  H1 is Subgroup of H2 & H3 is Subgroup of H4 implies [.H1,H3.] is
  Subgroup of [.H2,H4.]
proof
  assume H1 is Subgroup of H2 & H3 is Subgroup of H4;
  then commutators(H1,H3) c= commutators(H2,H4) by Th56;
  hence thesis by GROUP_4:32;
end;
