It is clear, however, that  excludes too much. There are many
situations, even trivial ones, of pairs (h,h') of equal allowable natural
transformations between shapes which are not both proper; for example,
1[AI], ^ A * 1 ^ : [A,I] -•[A,I]
bI(eSl)a"1, e(l$bA) : [A,I]B(AHI) -- I
where A is a non-constant shape.
Moreover, one can easily check that Proposition 7.6 of  can be
stated and proved without the hypothesis about propriety of shapes (this
is Proposition B of [18,§1]). In Proposition 7.8, however, propriety of
shapes is more essentially involved.
So, our work has started from a careful examination of the proof of
Proposition 7.8 of ,with respect to the point where the restriction
to "proper" shapes is essentially needed there. It is easily seen that
this point is in Case (ii) ([11,p.136]), where, for the inductive argu-
ment to work, the possibility that (in the notation of [11,p.136]) [B,C]
be associated via x with a prime factor of P must be excluded, and this
can be done only by introducing some special hypothesis.
In , this special hypothesis is the hypothesis that the shapes
In ,we take the class % of all the allowable graphs, which
are precisely of the form that (produces the case which) we want to ex-
clude, and we make the special hypothesis that the graphs of the natural
transformations in question are not in Wl , with lf[ being a class containing
7)f and defined so that it works well in the inductive proofs [more precise-
ly, lt[ is so defined, that the property "l%" of the graph of an allowable
natural transformation is inherited by the graphs of the allowable natural
transformations of lower rank, when we apply the cut elimination theorem].
With this hypothesis (replacing the hypothesis of  that the shapes are
proper), the method of  leads to the improved results of [18,§1],
In the following, we make the most direct (iQe. the weakest), with
respect to our purpose, special hypothesis: we suppose from the beginning
that, in the situation of Proposition 7.8 of  (which will be replaced
by Proposition 2 below), the allowable natural transformation h does not
admit the case we want to exclude; thus we form Proposition 2 below, which