How to understand the proof of Euclid's lemma?

Question

Mathematics

If a prime number p enters as a factor in the product ab, then it certainly enters as a factor either in a or in b. Indeed, if p were not a factor in either a or b, then by multiplying the factorizations into prime factors of the numbers a and b, we would obtain a factorization into prime factors of the number ab, which does not contain the factor p.
On the other hand , because it is assumed that p is a factor in the product ab, then this means that there is an integer t such that
ab = pt.
Therefore, multiplying p and the prime factorization of t, we obtain a prime factorization of ab containing the factor p.
Thus, we have to admit that there are two different factorizations of the number ab, and this contradicts the main theorem.

Question:
What properties does the number t have that the above actions lead to such a conclusion (to 2 different expansions)?
Let a = a1*a2 and b = b1*b2 (a1,a2,b1,b2 are prime numbers). p = a1, t = a2*b1*b2.
ab = pt, and indeed the expansion in the given cl is unique.

Answer 1

I.M. Vinogradov. Fundamentals of number theory. Page 20.