Polynomial-delay enumeration algorithms in set systems

We consider a set system (V,C⊆2V) on finite V of elements, where we call C∈C component. assume that two oracles L1 and L2 are available, given non-empty subsets X⊆Y⊆V, L1(X,Y) returns maximal component with X⊆C⊆Y; Y⊆V, L2(Y) all components C⊆Y. Given (V,C) along I items function σ:V→2I, is called solution if the common in C inclusively maximal; i.e., ⋂v∈Cσ(v)⊋⋂v∈Xσ(v) for any X∈C C⊊X. prove there exists an algorithm enumerating solutions (or components) delay bounded by polynomial respect to input size; upper bound number returned L2; running times oracles.