Algebra of Normal Forms Is

The articles [13], [4], [5], [2], [14], [3], [8], [6], [15], [9], [16], [10], [11], [12], [7], and [1] provide the notation and terminology for this paper. One can prove the following proposition (1) For all non-empty sets A, B, C and for every function f from A into B such that for every element x of A holds f(x) ∈ C holds f is a function from A into C. In the sequel A will be a non-empty set and a will be an element of A. Let us consider A, and let B, C be elements of FinA. Let us note that one can characterize the predicate B ⊆ C by the following (equivalent) condition: (Def.1) for every a such that a ∈ B holds a ∈ C. Let A be a non-empty set, and let B be a non-empty subset of A. Then B →֒ is a function from B into A. The following proposition is true (2) For every non-empty set A and for every non-empty subset B of A and for every element x of B holds ( B →֒ )(x) = x. In the sequel A denotes a set. Let us consider A. Let us assume that A is non-empty. The functor [A] yielding an non-empty set is defined by: