## purpose

Let us now take a quick look at the class $N$, with the purpose of determining how much of Theorems 1 and 2 is true here.

For this purpose, we first define ......

For our purposes here, the best way is to base the proof on the following theorem, a derivation of which can be found in [P].

The purpose of this paper is to expand substantially the class of maps for which the index can be computed.

A detailed exposition, more suited to the purposes of the present article, is given in [9].

The motivation for the results of this section is the following result of John (paraphrased slightly to suit our purposes).

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