We make the following provisional definition, which is neither general nor particularly elegant, but is convenient for the induction which is to follow.
With Lemma 2 in mind, we make the following ad hoc definition.
The preceding definitions can of course equally well be made with any field whatsoever in place of the complex field.
He proved the following theorem (see Section 2 for pertinent definitions).
The first of the above equalities is a matter of definition, and the second follows from (3).
Another way is to extend the definition of the index to closed curves by setting ......
Our definition agrees with the one in [3].
The definition is legitimate $\langle$correct$\rangle$, because ......
The precise definitions follow.
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