Infinite Primes Proof is complete ?

Hello ,

The proof to infinite prime numbers is saying that if you create new

number N from factoring the known finite prime series(P1*P2..*Pn) and

add 1 ,you will have number that will be new prime ,or be factored

from new prime that is not included in the finite prime series

My question is ,how do I prove that the N cannot be factored from

P1*P1*P2*... which one or more of the primes from the finite series appears more than

once,the proof is talking only about the case that P1,P2,..Pn appears

once in the factoring series but what about the option that it appears

more than once ? .

Thanks