  Fuzzy Rule Interpolation (FRI) reasoning methods have been introduced to
address sparse fuzzy rule bases and reduce complexity. The first FRI method was
the Koczy and Hirota (KH) proposed "Linear Interpolation". Besides, several
conditions and criteria have been suggested for unifying the common
requirements FRI methods have to satisfy. One of the most conditions is
restricted the fuzzy set of the conclusion must preserve a Piece-Wise Linearity
(PWL) if all antecedents and consequents of the fuzzy rules are preserving on
PWL sets at {\alpha}-cut levels. The KH FRI is one of FRI methods which cannot
satisfy this condition. Therefore, the goal of this paper is to investigate
equations and notations related to PWL property, which is aimed to highlight
the problematic properties of the KH FRI method to prove its efficiency with
PWL condition. In addition, this paper is focusing on constructing benchmark
examples to be a baseline for testing other FRI methods against situations that
are not satisfied with the linearity condition for KH FRI.
