 That is, the Skolem-Hull of \(e\) is the set \(e\) together with every \(v\)-witness of every formula over every \(e\)-assignment. Roughly, the idea is to start with \(e\) and then throw in enough elements to make each existentially quantified formula true. But we cannot rest content with the Skolem-hull, however. Once we throw the “witnesses” into the domain, we need to deal with \(sk(e)\) assignments. In effect, we need a set which is its own Skolem-hull, and also contains the given subset \(d_1\).