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I am focused on Section 7.2 Euclidean, Principal Ideal, Unique Factorization Domains ... ...

I need help with the proof of Theorem 7.2.14 ... ... Theorem 7.2.14 and its proof reads as follows:

View attachment 8275In the above proof by Bland we read the following:

"... ... But then \(\displaystyle a = apx' + aby' \), so \(\displaystyle p|apx'\) and \(\displaystyle p|aby'\). ... ... "Can someone please explain exactly how/why \(\displaystyle p|aby'\)... ... ?

Peter