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
  for N being strict normal Subgroup of G holds [.N,H.] is Subgroup of N
  & [.H,N.] is Subgroup of N
proof
  let N be strict normal Subgroup of G;
  now
    let a;
    assume a in [.N,H.];
    then consider F,I such that
A1: len F = len I and
A2: rng F c= commutators(N,H) and
A3: a = Product(F |^ I) by Th61;
    commutators(N,H) c= carr N by Th55;
    then rng F c= carr N by A2;
    then a in gr carr N by A1,A3,GROUP_4:28;
    hence a in N by GROUP_4:31;
  end;
  hence [.N,H.] is Subgroup of N by GROUP_2:58;
  now
    let a;
    assume a in [.H,N.];
    then consider F,I such that
A4: len F = len I and
A5: rng F c= commutators(H,N) and
A6: a = Product(F |^ I) by Th61;
    commutators(H,N) c= carr N by Th55;
    then rng F c= carr N by A5;
    then a in gr carr N by A4,A6,GROUP_4:28;
    hence a in N by GROUP_4:31;
  end;
  hence thesis by GROUP_2:58;
end;
