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 $\displaystyle f(x) \in F[X] $ a non-constant polynomial. Then

(1) $\displaystyle f(x) = u p_1(x) p_2(x) ... p_s(x) $, where $\displaystyle u \in F, u \neq 0 $, and each $\displaystyle p_i(x) $ 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 $\displaystyle f(x) = u p_1(x)p_2(x) ... p_s(x) = vq_1(x)q_2(x) ... q_t(x)$

where u,v $\displaystyle \in F $ and the $\displaystyle p_i(x) $ and $\displaystyle q_j(x) $ are monic and irreducible, then

$\displaystyle p_1(x) | q_1(x)q_2(x) ... q_r(x) $ ... etc etc

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Can someone show me (formally) why $\displaystyle p_1(x) | q_1(x)q_2(x) ... q_r(x) $ follows in the argument above

Peter