Corollary

If $N$ is finitely presented and $N_p$ is a free $A_p$-module of rank $p$, there exists $f \in A - p$ such that $N$, is a free $A_f$-module of rank $p$.

There exist by hypothesis $p$ elements $x_i \in \mathbf{N}$ ($1 \leq i \leq p$) such that the $x_i/1$ form a basis of the free $A_,-$-module $N_p$. Consider the homomorphism $u : A^p \to \mathbf{N}$ such that $u(e_i) = x_i$ for $1 \leq i \leq p$, $(e_i)_{1 \leq i \leq p}$ being the canonical basis of $A^p$. As $u_p$ is bijective by hypothesis, there exists $f \in A - p$ such that $u_f$ is bijective, by virtue of Proposition 2.