4

R.VOREADOU

g:B —- D and central x and y, is done as in [18], where we observe that

(now) the natural transformations, on which the induction hypothesis is

used, clearly satisfy the last hypothesis of the (present) proposition,

since h does so.

The case of h being of type IT is done as in [11]; that (in the nota-

tion of [11])

fa"1:

([Q,M]SP)S(NJSB) -^ C ,

on which the induction hypothesis is used, satisfies (as needed) the last

hypothesis of the present proposition, is proved as follows: Suppose fa

can be written in the form

([Q,M]HP)H(NHB) X" ([F,G]18E)HH

£"m

y GjaH

- 1 ^ C

where x", f", g" are allowable natural transformations,

xM

is central,

fM

cannot be written in the form and, via x", [Q,M] is associated with a

prime factor of E and [F,G] is associated with a prime factor of P. Let Pp

(resp.PH) be an iterated S-product of the prime factors of P associated

via xM with prime factors of E (resp.H). Similarly define Bp and BH. There

are obvious centrals

s:P -^ ([F,G]EPE)18PH , z:B-^BHSB£ , t:([Q,M]EPE)BBE -^ E

and

x" = ((lSt)Sl)u((lSs)18(lSz))

where u is the obvious central with domain

([Q,M]iS(([F,G]SPE)iSPH))S(Nia(BHSBE))

and codomain

([F,G]S(([Q,M]®PE)JSBE))B((PHEN)]SBH)

The mate under [ " (fa ) of any element of v(Bp) is the same as the mate un-

der rf" and is in v(M)+v(N)+v(B)+v(C) by the form of I " (fa" ) (as determined

by Th) and in v(E)+v(F) by the form of fa" (as displayed with f ), so

it must be in v(BF). Then the graph of the composite

-1

([Q,M]SPE)SBE — ^ E -^U~F — - FSI

is of the form f*Ky for graphs JLL:[Q,M]SPF—• F , v:Bp— ~ I . By Proposition

B of [18], b ft = mSn for allowable natural transformations n:Bp—v I

and m:[Q,M]SPE-^ F . So ft = b(mSn). (Note that this equality also holds

in case there are no prime factors of B associated with prime factors of E

via xM, when B£ = I.) Then