  The ``Fundamental Theorem" given by Arnold in [2] asserts the persistence of
full dimensional invariant tori for 2-scale Hamiltonian systems. However,
persistence in multi-scale systems is much more complicated and difficult. In
this paper, we explore the persistence of lower dimensional invariant tori for
multi-scale Hamiltonian systems, which play an important role in dynamics of
resonant Hamiltonian systems. Moreover, using the corresponding results we give
a quasi-periodic Poincar\'{e} Theorem for multi-scale Hamiltonian systems,
i.e., at least $2^{m_0}$ families resonant tori survive small perturbations,
where the integer $m_0$ is the multiplicity of resonance.
