Common zeroes of multivariable p-adic polynomials

I am reading Serre's A Course in Arithmetic, and I have some doubts about a proposition regarding the set of common zeroes of polynomials with $p$-adic coefficients. In the following, note that $\mathbf Z_p$ denotes the $p$-adic integers (not $\mathbf Z/p\mathbf Z$), and $A_n$ denotes $\mathbf Z/p^n\mathbf Z$. (Hence, $\mathbf Z_p=\varprojlim A_n$.) The following is the relevant excerpt from the text.

I think I understand most of this, except in the statement of proposition 5 under (ii), why is it that we require $n>1$ instead of $n\geq 1$? I don't seem to understand where the proof requires the fact that $n\neq 1$. Is this a typo or am I missing something obvious? Thanks in advance.

Just to get this off the unanswered list...

As mentioned by Mindlack in the comments, the two versions of (ii) with $n\geq 1$ or $n>1$ are equivalent. One sees this by noting that a zero of a polynomial mod $p^2$ also gives a zero mod $p$ for obvious reasons. In the later pages, Serre also states similar results with $n>1$, and they are also equivalent to the $n\geq 1$ case for the same reason.