theorem Th55:
  for N being strict normal Subgroup of G holds commutators(H,N)
  c= carr N & commutators(N,H) c= carr N
proof
  let N be strict normal Subgroup of G;
  thus commutators(H,N) c= carr N
  proof
    let x be object;
    assume x in commutators(H,N);
    then consider a,b such that
A1: x = [.a,b.] and
    a in H and
A2: b in N by Th52;
    b" in N by A2,GROUP_2:51;
    then b" |^ a in N |^ a by GROUP_3:58;
    then b" |^ a in N by GROUP_3:def 13;
    then x in N by A1,A2,GROUP_2:50;
    hence thesis by STRUCT_0:def 5;
  end;
  let x be object;
  assume x in commutators(N,H);
  then consider a,b such that
A3: x = [.a,b.] and
A4: a in N and
  b in H by Th52;
  a |^ b in N |^ b by A4,GROUP_3:58;
  then
A5: a |^ b in N by GROUP_3:def 13;
  x = a" * (a |^ b) & a" in N by A3,A4,Th16,GROUP_2:51;
  then x in N by A5,GROUP_2:50;
  hence thesis by STRUCT_0:def 5;
end;
