COHERENCE AND NON-COMMUTATIVE DIAGRAMS IN CLOSED CATEGORIES 11
h(x') = q.(pHl) with Tp = [f and Tq = Tg'. By induction, (p,f')^ty im-
plies p = f! , and (q,gT)^3f implies q = g*. So, using K5 and 1H2 as in a
preceding case, we find that (h,h')^^ implies (now) h = h'. Subcases II,
III and IV lead to h(xf)~ being of type ® (see [18];we arrive at this be-
fore using the special hypothesis '7#l" of [18]), so they are excluded by
the present hypothesis that h is not of type K.
Case (ii). [Q,M] is associated with a prime factor of A via x.
Subcase I: [B,C] associated with a prime factor of P via x. By our assump-
tion and Lemma 3, Tf and Tf cannot be written in the form ; then, by
Lemma 4, Q and B are non-constant. Then, since (neither h nor h' and there-
fore) neither h(x') nor hT(xT)~ is of any one of the forms "central", $
or 7 f , we have (h(xf) " 1,h! (x») " 1) e WQ^W which, by T2 , implies (h,hT)e^;
a contradiction. So this subcase is excluded.
Subcase II: [B,C] associated with a prime factor of N via x. Working as in
Subcase II of Case (ii) in the proof of Proposition C of [18],we find
that this subcase is excluded since (in the notation of [18]), if r(Pn)
= 0 , then ft (of [11],as in [18]) satisfies the hypotheses of Proposi-
tion 2, because f cannot be written in the form , and therefore, by Prop-
osition 2, ft is of type and hence so is f, a contradiction, and, if
r(Pn)0, then h is of type B (we arrive at this in [18] before using the
special hypothesis 'V^M ), another contradiction.
Case (iii). [Q,M] is associated with a prime factor of D via x.
Subcase I: [B,C] associated with a prime factor of P via x. By Lemma 3, Tf
and rff cannot be written in the form . So, working as in Subcase I of
Case (iii) in the proof of Proposition C of [18],we find that h is of
type S, so this subcase is excluded.
Subcase II: [B,C] associated with a prime factor of N via x. With the nota-
tion of [18] for this subcase of Case (iii) in the proof of Proposition C,
we find that the case r(P.)0 is excluded since it leads to h being of
type S. In case r(P.) = 0, we continue as in [11] (the respective subcase
in the proof of Proposition 7.8) until we reach the composite g(lSt)w. In
order to get g(lSt)w = r(pBl) for allowable natural transformations
p:P—- Q and r:MS(CSR) —- S, which in the corresponding case in the proof
of Proposition C was given by the induction hypothesis, we apply Proposi-
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