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

.