[vladsotirov]

You've correctly shown that "f2 does not contain p". However, the subtle bit of reasoning "f2 cannot be evenly divisible by p as that would require it to have a prime factor of p" is not justified by any of your assumptions.

Let me be clear: f2 is a fixed product of primes p_1, p_2, p_3, etc. It is certainly true that each p_i divides evenly f2. What you use, however, is the converse claim: that every prime p that divides f2 is on this list.

This converse claim, that "every prime p that divides f2 is on the list of primes that we built f2 out of", does not follow from "no number of primes can be multiplied together to make another prime". The latter implies only that a prime that both divides f2 and can be built out from the given list of primes, has to be on the list of primes.

In fact, the converse claim "every prime p that divides f2 is on the list of primes that we built f2 out of" is essentially one variant of the statement of unique prime factorization, so you actually fell into the trap of circular reasoning.