(4) ∅ is Subset of X,

Summary. The text includes theorems concerning properties of subsets, and some operations on sets. The functions yielding improper subsets of a set, i.e. the empty set and the set itself are introduced. Functions and predicates introduced for sets are redefined. Some theorems about enumerated sets are proved. The articles [2], [3], and [1] provide the terminology and notation for this paper. In the sequel E, X denote objects of the type set; x denotes an object of the type Any. One can prove the following propositions: (1) E � = ∅ implies (x is Element of E iff x ∈ E), (2) x ∈ E implies x is Element of E, (3) X is Subset of E iff X ⊆ E. We now define two new functors. Let us consider E. The functo