Computations with parameters: a theoretical background

Consider a computational algebraic problem with the inputs depending on parameters. The aim of the computation is to stratify the variety of parameters such that for each stratum $\mathcal W_\alpha$ the solution of the problem for arbitrary values of parameters from each stratum as function of parameters is computed by the same algebraic formulas depending only on the stratum $\mathcal W_\alpha$.
We suggest a model for computations with parameters which is convenient for practical aims and prove a fundamental result for it.