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) $\displaystyle \neq $ 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) $\displaystyle \leq $ deg g(x)

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

Suppose we have:

f(x) = $\displaystyle q_1 (x)g(x) + r_1(x) $

f(x) = $\displaystyle q_2 (x)g(x) + r_2(x) $

where $\displaystyle r_i(x) = 0 $ or deg $\displaystyle r_i(x) \leq $ deg g(x).

Subtracting the two equations give we get

0 = $\displaystyle [q_1(x) - q_2(x)] g(x) + [r_1(x) - r_2(x)]$

Hence

(3) $\displaystyle [q_1(x) - q_2(x)]g(x) = r_2(x) - r_1(x) $

We must have $\displaystyle r_2(x) - r_1(x) = 0 $, for otherwise deg $\displaystyle [r_2(x) - r_1(x)] \leq $ 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 "$\displaystyle r_2(x) - r_1(x) = 0 $, for otherwise deg $\displaystyle [r_2(x) - r_1(x)] \leq $ deg g(x) which would makje equation (3) impossible " follows?