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 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;
