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;
