theorem Th10:
  B is Subgroup of A implies commutators(A,B) c= carr A
proof
  assume
A1: B is Subgroup of A;
  let x be object;
  assume x in commutators(A,B);
  then consider a,b such that
A2: x = [.a,b.] & a in A & b in B by GROUP_5:52;
A3: b in A by A1,A2,GROUP_2:40;
then A4: a * b in A by A2,GROUP_2:50;
  a" in A & b" in A by A2,A3,GROUP_2:51;
  then a" * b" in A by GROUP_2:50;
  then
A5: (a" * b") * (a * b) in A by A4,GROUP_2:50;
  [.a,b.] = (a" * b") * (a * b) by GROUP_1:def 3;
  hence thesis by A2,A5,STRUCT_0:def 5;
end;
