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((1).G,H) = {1_G} & commutators(H,(1).G) = {1_G}
proof
A1: commutators((1).G,H) c= {1_G}
  proof
    let x be object;
    assume x in commutators((1).G,H);
    then consider a,b such that
A2: x = [.a,b.] and
A3: a in (1).G and
    b in H by Th52;
    a = 1_G by A3,Th1;
    then x = 1_G by A2,Th19;
    hence thesis by TARSKI:def 1;
  end;
  1_G in commutators((1).G,H) by Th53;
  hence commutators((1).G,H) = {1_G} by A1,ZFMISC_1:33;
  thus commutators(H,(1).G) c= {1_G}
  proof
    let x be object;
    assume x in commutators(H,(1).G);
    then consider a,b such that
A4: x = [.a,b.] and
    a in H and
A5: b in (1).G by Th52;
    b = 1_G by A5,Th1;
    then x = 1_G by A4,Th19;
    hence thesis by TARSKI:def 1;
  end;
  1_G in commutators(H,(1).G) by Th53;
  hence thesis by ZFMISC_1:31;
end;
