  In another paper with the same name\cite{frame}, we proposed a new
representation of Game Theory, but most results are given by specific examples
and argument. In this paper, we try to prove the conclusions as far as we can,
including a proof of equivalence between the new representation and the
traditional Game Theory, and a proof of Classical Nash Theorem in the new
representation. And it also gives manipulation definition of quantum game and a
proof of the equivalence between this definition and the general abstract
representation. A Quantum Nash Proposition is proposed but without a general
proof. Then, some comparison between Nash Equilibrium (NE) and the
pseudo-dynamical equilibrium (PDE) is discussed. At last, we investigate the
possibility that whether such representation leads to truly Quantum Game, and
whether such a new representation is helpful to Classical Game, as an answer to
the questions in \cite{enk}. Some discussion on continuous-strategy games are
also included.
