On the downward Löwenheim-Skolem Theorem for elementary submodels

We introduce a new definition of a model for a formal mathematical system. The definition is based upon the substitution in the formal systems, which allows a purely algebraic approach to model theory. This is very suitable for applications due to a general syntax used in the formal systems. For our models we present a new proof of the downward Löwenheim-Skolem Theorem for elementary submodels.