theorem Th22:
  a * b = a iff a + b = b + a
proof
  thus a * b = a implies a + b = b + a
  proof
    assume a * b = a;
    then a = (-b) + (a + b) by RLVECT_1:def 3;
    hence thesis by Th12;
  end;
  assume a + b = b + a;
  then a = (-b) + (a + b) by Th12;
  hence thesis by RLVECT_1:def 3;
end;
