reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

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;
