In this paper, a quantifier elimination method for positive real theory with order by Volker Weispfenning is extended to a decision procedure, from which a new and finer decision method for eleminentary theory of real addition with order is given. It is proved that its subclasses of fixed number of quantifiers is in correspondant polynomial hierarchy. The result of this paper is essentially an extension of the claim on positive real theory by N. Megiddo to the whole real theory, which is relatively simply enduced in this paper, and comparably better than the similar result by E.D. Sontag.