reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  commutators({} the carrier of G,A) = {} & commutators(A,{} the carrier
  of G) = {}
proof
  commutators({} the carrier of G,A) c= {}
  proof
    let x be object;
    assume x in commutators({} the carrier of G,A);
    then ex a,b st x = [.a,b.] & a in {} the carrier of G & b in A;
    hence thesis;
  end;
  hence commutators({} the carrier of G,A) = {};
  thus commutators(A,{} the carrier of G) c= {}
  proof
    let x be object;
    assume x in commutators(A,{} the carrier of G);
    then ex a,b st x = [.a,b.] & a in A & b in {} the carrier of G;
    hence thesis;
  end;
  thus thesis;
end;
