12
R.VOREADOU
tion 2. Clearly, g(l®t)w satisfies all the hypotheses of Proposition 2, ex-
cept, possibly, for the last one. If it does not satisfy the last hypothe-
sis of Proposition 2, then, working as in the last part of the proof of
Proposition 2 (but with h(x,)~ in the role of the h in that proof), we
find that h(x')~ is of the form
([Q,M]SP)SN -^—^- ([F,G]JaEf)KH' f*B1 GBH' g* S ,
where x* is central, f* and g* are allowable, f* cannot be written in the
form and, via x*, [Q,M] is associated with a prime factor of E1 and
[F,G] is associated with a prime factor of P. It is obvious that no prime
factor of E' is constant. By Lemmas 3 and 4, Tf* and Tf are not of type
and Q and F are non-constant. Then h = g*(f*$l)x*xT and, clearly,
[Q,M] is associated with a prime factor of E1 via x*xT(x') = x* and [F,G]
is associated with a prime factor of P via x'(x*xT) . Since neither h nor
hT can be written in any of the forms "central", 8 or ir , it follows that
(hjh'JeW1 , so (h,hf)£W, a contradiction. So g(lKt)w satisfies the last
hypothesis of Proposition 2. So, by Proposition 2, g(lKt)w = r(pKl) and
then h(x')"1 = q(p®l) with Tp = Tf! and Tq = Tg'; by induction, (p,f)^
implies p = ff and (qjg')^ implies q=g'; so, using J12 and 0^5 as in a
similar previous situation, from the hypothesis (h,h')^3V we get h = hf.
It is clear that (h,hf)£ JfQ implies |~ h = Th'e 1R0 , and (h,hf)e^ im-
plies Th = Th'eTJf, for the classes % and Tfl of [18,§1] (also given here
in §1 of Part Two). So Th = ThV^ implies (h,h')^, and therefore Theo-
rem 2 of [18] is a corollary of Theorem 5 proved above. However, Theorem 5
is of theoretical value only, until we give a finite test for deciding
whether (h,h')^9f . This will be done by using the results of Part TwoQ
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