Exercise 9.3. Show that the fuzzy systems in the form of (9.1) have the uni- versal approximation property in Theorem 9.1. Attachment: Lemma 9.1. Suppose that the fuzzy set B in (7.1) is normal with center Then the fuzzy systems with fuzzy rule base (7.1), product inference engine (7.23), singleton fuzzifier (8.1), and center average defuzzifier (8.18) are of the following form: where E U C Rn is the input to the fuzzy system, and f(x) V c R is the output of the fuzzy system Theorem 9.1 (Universal Approximation Theorem). Suppose that the input universe of discourse is a compact set in R". Then, for any given real continuous function g(x) on U and arbitrary e> 0, there exists a fuzzy system f(x) in the form of (9.6) such that sup lf (z) g(x)l