Exercise 30, § 1

Let $p$ be a prime number. Let $A$ be a filtered ring and $(J_r)_{r \in \mathbf{Z}}$ its filtration. Suppose that one has $J_0 = A$ and $p.1_A \in J_1$. Let $a$ and $b$ be elements of $A^J$ and $r \in \mathbf{N}$. a) If one has $a_d \equiv b_d \mod. J_r$ for all $d \in J_n$, then one has $\Phi_n(a) \equiv \Phi_n(b) \mod. J_{r+k}$ where $k = w_p(n)$. b) Suppose that, for every integer $m \geq 1$ and every $x \in A$, the relation $p.x \in J_{m+1}$ implies $x \in J_m$. If one has $\Phi_d(a) \equiv \Phi_d(b) \mod. J_{r+w_p(d)}$, for all $d \in J_n$, then $a_d \equiv b_d \mod. J_r$ for all $d \in J_n$.