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As Sharwyn sleeps, the polygons comprising many of her brain constructs process the memories and emotions she experienced during her waking hours. If these experiences are of minor significance then the polygons will process the information pertaining to these experiences by continually computing ever larger domain values for polynomial rational functions whose leading terms, as in f(x) = (ax^n + ...) / (bx^m + ...) will have exponents m>n. The domain values therefore get very large, very quickly as almost all of Sharwyn's (and most anyone else's) waking hours are filled with trivial experiences that do not initiate any emotional response. So as the limit of the domain values of these functions goes to infinity, with m>n, then the range values become very, very small until they become virtually zero. None of these experiences will be retained in Sharwin's medium or long-term memory, and those that might be retained in her short-term memory whilst she's awake will not survive sleep. If her waking experiences are of some importance, such as something that causes her surprise or makes her quite emotional, then during sleep these experiences will be processed for subsequent retrieval from her medium-term memory. In this instance polygons will process these fairly important experiences by continually computing ever larger domain values of PRFs whose leading terms will have m = n. As the limit of the domain values of these functions goes to infinity with m = n, the range values become ever closer to the ratio of the leading term coefficients a/b. If she suffers a life-threatening experience or something that causes great emotional joy or upset, then during sleep her polygons will process these experiences for subsequent retrieval from her long-term memory. In this instance polygons will process these very important experiences by continually computing ever larger domain values of PRFs whose leading terms will have n>m. As the limit of the domain values of these functions goes to infinity with n>m, the range values become ever closer to infinity (if the experiences are pleasurable) or negative infinity (if the experiences are disturbing). As Sharwyn is having a nightmare this means that many of her brain polygons are processing memories of something that has caused her great upset (the infidelity of her lover) by using PRFs with leading terms n>m and range values approaching negative infinity.
The procedure for breaking Sharwyn's nightmare requires that Zanzi undertake high-resolution analysis of all those PRFs with leading terms n>m and range values approaching negative infinity that are the cause of the nightmare. Zanzi then formulates an almost infinite number of functions to intersect the n>m PRFs, such that each point of intersection is undefined for the intersecting function. The result is that a hole exists at every point of the coordinate domain and range pairs generated by the PFRs, and these holes nullify the PFRs’ ability to negatively affect Sharwyn’s psychology while she sleeps, thus ending her nightmare. At the same time, Zanzi's music and implanted images of Selune create PRFs with leading terms n>m and range values approaching positive infinity, so providing Sharwyn extremely pleasurable experiences while she dreams – too pleasurable in the end, as they caused her to wake up quite quickly.