Proof
(related to Lemma: Generalized Euclidean Lemma)
 By hypothesis, the prime number \(p\) divides a product of integers \(n=\prod_{i=1}^\rho n_i.\)
 Assume, $p\not\mid n_i$ for all $i=1,\ldots,\rho.$
 Then, according to coprime primes, $\gcd(p,n_i)=1$ for all $i=1,\ldots,\rho.$
 Therefore, according to divisors of a product coprime to factors it follows that $p\not\mid n.$
 This is a contradiction to the hypothesis.
 Therefore, $p\mid n_i$ for at least one $i\in\{1,\ldots,\rho\}.$
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References
Bibliography
 Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927