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**Deveno** the a_{i} (in the first part) are completely arbitrary elements. he uses a chain built from n+1 of them to show that 2 must be equal (since we can only have a chain of length n that is STRICTLY decreasing).

on the other hand, from a given strictly decreasing chain of length n (assumed to exist from the outset, and there may be several such), he constructs a word of length n, showing (from the first part) that the nilpotency is both less than n+1, and at least n, and therefore n.

i assume that by 0 he means "absorbing element" (i.e. z: az = z for all a in S), like for example the 0 map of the semigroup End(V), for a vector space V.