theorem Th11:
  B is Subgroup of A implies [.A,B.] is Subgroup of A
proof
A1: gr carr A is Subgroup of A by Lm1;
  assume B is Subgroup of A;
  then commutators(A,B) c= carr A by Th10;
  then [.A,B.] is Subgroup of gr carr A by GROUP_4:32;
  hence thesis by A1,GROUP_2:56;
end;
