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proof of inifinite primes

Could someone help me understand this proof?

It says, by assumption, for some .

It also says and by assumption, . It makes perfect sense if the in the first statement and the in the second are different values of . But it declines this by saying that (the is the same in both statements). I don't see how this proves anything however, because you could simply choose a different prime for a+1, instead of sticking with the same prime and forcing a contradiction. Also I'm not confortable with all the assumptions being made.

N.B: Exercise 3.2 states that

and Axiom 3 states that .