  The concept of variety with IBN (invariant basic number) propriety first
appeared in ring theory. But we can define this concept for arbitrary variety
of universal algebras with arbitrary signature; see Definition 1.4. The proving
of the IBN propriety of some variety is very important in universal algebraic
geometry. This is a milestone in the study of the relation between geometric
and automorphic equivalences of algebras of this variety. In this paper we
prove very simple but very useful for studying of IBN proprieties of different
varieties Theorems 2.1 and 3.2. We will consider applications of this theorem.
We will consider many-sorted universal algebras as well as one-sorted. So all
concepts and all results will by generalized for the many-sorted case.
