> f2 does not contain p, and we cannot construct a prime p from other primes or composites. This means that f2 cannot be evenly divisible by p, as that would require it to have a prime factor of p which it does not.

No it doesn't. Suppose p, m, q and r are (distinct) primes where f1 = p * m, f2 = q * r. Then why can't you have (q * r) / p = m? The fundamental theorem of arithmetic says you can't, but you are trying to prove that.