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Case 3
Next we study the case that
and
is odd.
In this subsection we use methods that are very similar to the one used in Subsection 5.4.2 and Subsection 5.4.3, and hence we omit the detail of the argument.
In Fig. 5.21 we suppose that Predictions 5.1, 5.2, 5.3,5.4 and 5.5 are valid for
and for
. We also suppose that these predictions are valid for
and
. Under these assumptions we prove that
.
For
by Prediction 5.1 the first number from the right end of the row is
.
We suppose that

(5.51) 
and

(5.52) 
We also assume that
and 
(5.53) 
If we are to prove that
, by Lemma 5.1 we have only to prove that
is the smallest number that does not belong to
.
.
We have
.
.
Therefore we have
.
The case that
can be treated in the similar way, so we omit it.
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