I am reading Papantonopoulou - Algebra, Ch 8, Rings of Polynomials

Theorem 8.2.2 (Division Algorithm) reads as follows: (see attachment for full proof)

Let F be a field and f(x) and g(x) elements of F[x], with g(x) 0 . Then there existuniqueelements Q(x) and r(x) of F[x] such that

(1) f(x) = q(x)g(x) + r(x)

(2) r(x) = 0 or deg r(x) deg g(x)

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I am having trouble following the uniqueness argument which runs as follows:

Suppose we have:

f(x) =

f(x) =

where or deg deg g(x).

Subtracting the two equations give we get

0 =

Hence

(3)

We must have , for otherwise deg deg g(x) which would makje equation (3) impossible .... etc etc

(see the proof on the attachment)

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Can anyone show me (formally) why the last statement ie " , for otherwise deg deg g(x) which would makje equation (3) impossible " follows?