theorem
  commutators({a},{b}) = {[.a,b.]}
proof
  thus commutators({a},{b}) c= {[.a,b.]}
  proof
    let x be object;
    assume x in commutators({a},{b});
    then consider c,d such that
A1: x = [.c,d.] and
A2: c in {a} & d in {b};
    c = a & b = d by A2,TARSKI:def 1;
    hence thesis by A1,TARSKI:def 1;
  end;
  let x be object;
  assume x in {[.a,b.]};
  then
A3: x = [.a,b.] by TARSKI:def 1;
  a in {a} & b in {b} by TARSKI:def 1;
  hence thesis by A3;
end;
