We must now bring dependence on $d$ into the arguments of [9].

The proof will be divided into a sequence of lemmas.

We can factor $g$ into a product of irreducible elements.

Other types fit into this pattern as well.

This norm makes $X$ into a Banach space.

We regard (1) as a mapping of $S^2$ into $S^2$, with the obvious conventions concerning the point $\infty$.

We can partition $[0,1]$ into $n$ intervals by taking ......

The map $F$ can be put $\langle$brought$\rangle$ into this form by setting ......

The problem one runs into, however, is that $f$ need not be ......

But if we argue as in (5), we run into the integral ......, which is meaningless as it stands.

Thus $N$ separates $M$ into two disjoint parts.

Now (1) splits into the pair of equations ......

Substitute this value of $z$ into $\langle$in$\rangle$ (7) to obtain ......

Replacement of $z$ by $1/z$ transforms (4) into (5).

This can be translated into the language of differential forms.

Implementation is the task of turning an algorithm into a computer program.

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