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**DivideBy0** Could someone help me understand this proof?

It says, by assumption, $\displaystyle p=p_i$ for some $\displaystyle i=1,2,3,...,n$.

It also says $\displaystyle p_i | a$ and by assumption, $\displaystyle p_i | a+1$. It makes perfect sense if the $\displaystyle p_i$ in the first statement and the $\displaystyle p_i$ in the second are different values of $\displaystyle i$. But it declines this by saying that $\displaystyle p_i | (a+1)-a=1$ (the $\displaystyle p_i$ 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 $\displaystyle n|a \wedge n|b \Rightarrow n|a-b$

and Axiom 3 states that $\displaystyle n|1 \Rightarrow n=\pm 1$.