Proposition 1

Let A be a filtered ring and $(\mathbf{J}_n)_{n \in \mathbf{Z}}$ its filtration. Assume that one has $\mathbf{J}_0 = \mathbf{A}$ and $p.1_A \in \mathbf{J}_1$. Let m and n be integers such that $m \geqslant 1$ and $n \geqslant 0$, and let $a_0, ..., a_n, b_0, ..., b_n$ be elements of A.

a) If one has $a_i \equiv b_i \mod \mathbf{J}_m$ for $0 \leqslant i \leqslant n$, then one has

b) Suppose that, for every integer $k \geqslant 1$, and every $x \in \mathbf{A}$, the relation $p.x \in \mathbf{J}_{k+1}$ entails $x \in \mathbf{J}_k$. If one has $\Phi_i(a_0, ..., a_i) \equiv \Phi_i(b_0, ..., b_i) \mod \mathbf{J}_{m+i}$ for $0 \leqslant i \leqslant n$, then one has $a_i \equiv b_i \mod \mathbf{J}_m$ for $0 \leqslant i \leqslant n$.