Proof that there are infinitely many primes of the form $4n+1$

Question

After proving this

The author says this

But if I let $q = 2^2.3.5....p + 1$ instead, and use theorem $11$, then doesn't that prove that there are infinitely many primes of the form $4n + 1$?

Answer 1

No, it does not. $q$ could still be a product of all primes that are of the form $4n + 3$.

Answer 2

Notice $(4a+3)(4b+3)$ is of form $4t+1$.

So a number of form $4t+1$ can be made by multiplying only the primes of form $4a+3$