Abstract. The results established in this paper are in connection with the Relative Internal Set Theory (R.I.S.T.). The main result is the general principle of choice: Let be a level and let be an external bounded formula of the language of R.I.S.T.. Suppose that to each element , dominated by , corresponds an element such that holds, then there exists a function of choice such that, which is a very general principle of choice, for every dominated by , holds. More than that, we establish that if all the elements are uniformly dominated by a level then we can prescribe that the function of choice is also dominated by .