I am reading Papantonopoulou - Algebra - Ch 8 Rings of Polymonials

On page 251 (see attachment) Papantonopoulou states the Unique Factorization Theorem as follows:

Let F be a field and a non-constant polynomial. Then

(1) , where , and each is a monic irreducible polynomial over F

(2) Except for the order of the irreducible facotrs, the factorization in (1) is unique

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The proof of (1) is fine, but I am having trouble with the start of the uniqueness proof which reads as follows: (see attachment)

Use induction on deg f(x) = n.

If n=1, uniqueness holds.

So now assume uniqueness holds for all polynomials of degree < n = deg f(x).

If

where u,v and the and are monic and irreducible, then

... etc etc

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Can someone show me (formally) why follows in the argument above

Peter