Essential variables and positions in terms

The paper deals with $\mathrm{\Sigma }-$composition of terms, which allows us to extend the derivation rules in formal deduction of identities. The concept of essential variables and essential positions of terms with respect to a set of identities is a key step in the simplification of the process of formal deduction. $\mathrm{\Sigma }-$composition of terms is defined as replacement between $\mathrm{\Sigma }$-equal terms. This composition induces $\Sigma R-$deductively closed sets of identities. In analogy to balanced identities we introduce and investigate $\mathrm{\Sigma }-$balanced identities for a given set of identities $\mathrm{\Sigma }$.