theorem Th58:
  x in commutators G iff ex a,b st x = [.a,b.]
proof
  thus x in commutators G implies ex a,b st x = [.a,b.]
  proof
    assume x in commutators G;
    then ex a,b st x = [.a,b.] & a in (Omega).G & b in (Omega).G by Th52;
    hence thesis;
  end;
  given a,b such that
A1: x = [.a,b.];
  thus thesis by A1;
end;
