 The next higher level in EH are K-systems. Unlike in the cases of ergodic and mixing systems, there is unfortunately no intuitive way of explaining the standard definition of such systems, and the definition is such that one cannot read off from it the characteristics of K-systems (we state this definition in the  Appendix,  Section C). The least unintuitive way to present K-systems is via a theorem due to Cornfeld et al (1982, 283), who prove that a dynamical system is a K-system iff it is K-mixing. A system is K-mixing iff for any subsets \(A_0, A_1 , \ldots ,A_r\) of \(X\) (where \(r\) is a natural number of your choice) the following condition holds: