Index sets occurring in indexing terms can be themselves indexed. (We have already seen
an example above in set of sets.) Let us define an index set I and based on it an
indexed index set J = {J_{i}} where J_{i} is an index set and i ∈ I. The elements of J are
index sets themselves. In this case the index mechanism is not applied to a singleton
but to a set. As an example, suppose I = {a,b}, J_{a} = {1, 2}, and J_{b} = {2, 3} then
J = {{1, 2},{2, 3}}. Based on these two sets I and J, one can form the following indexed
notation:

(7) |

It is important to note, however, that the index i in i ∈ I is an active index, whereas the i in
J_{i} is a passive index. Now (7) can be interpreted as follows:

(8) |

This means that the concept of indexed index set can be reduced to compound index set and does not add any new features. Nevertheless, a notation like (7) may sometimes be more convenient, because the tuples are presented as a hierarchical structure instead of a flat list.