theorem
  G is commutative Group iff commutators G = {1_G}
proof
  thus G is commutative Group implies commutators G = {1_G} by Th57;
  assume
A1: commutators G = {1_G};
  G is commutative
  proof
    let a,b;
    [.a,b.] in {1_G} by A1;
    then [.a,b.] = 1_G by TARSKI:def 1;
    hence thesis by Th36;
  end;
  hence thesis;
end;
