We easily check that, for every central natural transformation u:S —-T ,
the shapes S and T are made up of the same prime factors and the (central)
graph |~u:S —-T associates each prime factor of S with its copy in T, in
the sense that, via Tu, the variables of any prime factor K of S are link-
ed with the variables of the copy of K in T in an order preserving way;
then we also say that the prime factors are associated via u (instead of
saying via |~u) .
An iterated S-product of prime shapes X.,,Xo,...,X is a shape T
which is gotten by the substitution of the shapes X-.,X~,...,X for the va-
riables in some integral shape D, where D has precisely n variables and
involves no I ; then the prime factors of T are precisely the shapes X.
The iterated S-product of the members of an empty collection of
prime shapes is, by definition, the shape I .
We note that the arguments of  have been modified and used in
, where a general cut elimination theorem is proved. As a special case
of that theorem we get the fact that the class of allowable arrows of #"
(p.xi) [i.e. the class Arr^ of the (abstract) allowable natural transfor-
mations between shapes] has the cut elimination property given on p.xiv0
As a consequence of this fact, all the arguments and results of  and
, which use the cut elimination property and are given for allowable
natural transformations with respect to some closed category V , are also
valid for (abstract) allowable natural transformations between shapes, and
in this sense they will be used in the present paper.
This completes the general introduction. Most of the references to
 that will appear in the text and have not already been explained here,
are related with details of proofs. For details of certain proofs, refer-
ences to  are also given.
Thanks are extended to Professor S.MacLane for his interest in all
phases of this research, his questions and advice; also, to the referee
for pointing out certain weaknesses of the first version.