  In this work, we present the equivalent of many theorems available for
continuous time systems. In particular, the theory is applied to Averaging
Theory and Separation of time scales. In particular the proofs developed for
Averaging Theory and Separation of time scales departs from those typically
used in continuous time systems that are based on twice differentiable change
of variables and the multiple use of the Implicit Function Theorem and Mean
Value Theorem. More specifically, by constructing a suitable Lyapunov function
only Lipschitz conditions are necessary. Finally, it is shown that under mild
condition on the so-called "interconnection conditions" the proposed tools can
guarantee semi-global exponential stability rather than the more stringent
local exponential stability typically found in the literature
