A method for testing and zero-order integral minimal realization of threshold functions using a reduced set of the total number of inequalities is suggested in the present paper. The paper utilizes the concept of essential second-order incremental weights and other basic ideas developed in an earlier paper by the authors (Choudhury et al. 1966). It is first shown that the information regarding the essentiality of the second-order incremental weights and some of the ordering relations necessary for testing linear separability of a function can be obtained from a knowledge of the ordering relations existing between only some specific pairs of coefficient combinations called the standard test set, which relations can, in turn, be readily obtained from a consideration of only a sub-set of the set of total number of inequalities of the function. On the basis of this, it is next shown that a zero-order integral minimal realization of the function can be obtained by using only a reduced set of the total number of inequalities.