Abstract
A proof of the following theorem is given: If M \mathcal {M} is a finitely generated extension of a partial difference field K \mathcal {K} then every subextension of M / K \mathcal {M}/\mathcal {K} is finitely generated. An integral measure of partial difference field extensions having properties similar to the dimension of field extensions and the limit degree of ordinary difference field extensions and a new method of computing transformal transcendence degree are developed.
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