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In this section we prove that an open neighborhood of is contained in .
### Lemma 3.1

If is compact, , **|z| < 1**, and
then every element of **B** is expressible in the form
In particular,
if , then .

### [Proof]

### Proposition 3.1

If , ,
then .

### Lemma 3.2

If is compact,
, ,
and
where int denotes the interior of **S**,
then there is a neighborhood **N** of **x** such that

### Lemma 3.3

If for some ,
, **|x| =1** and
then

### Proposition 3.2

If **|x| =1**, , then .

Proving is trickier,
because it will not do to take **B** as a disc of radius if
is small compared to .
We will instead take **B** as a parallelogram that becomes flatter and flatter as
.
The following two lemmas will be used in verifying the condition of Lemma 3.1.

### Lemma 3.4

Let .
Let .
Then for
contains .

### Lemma 3.5

Let **T**, be as in Lemma 3.4.
Let **B** be the square with vertices .
Then for ,

### Proposition 3.3

.

We now combine all the results of this section.

### Theorem 3.1

There is an open neighborhood of contained in .

### Corollary 3.1

If for sufficiently small then **z** is a multiple zero of some **0,1** power series.

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