Proposition 3

Let A, B, C be three local rings and $u : A \to B, v : B \to C$ two inflations. Then $v \circ u$ is an inflation. Let $(B_\lambda)_{\lambda \in \Lambda}$ and $(C_\mu)_{\mu \in M}$ be families of sub-A-algebras of B and of sub-B-algebras of C respectively, having properties a), b), c) of Def. 2. On the set N sum of $\Lambda$ and M, consider the order relation inducing on $\Lambda$ and M the given orders and such that one has $\lambda < \mu$ for $\lambda \in \Lambda, \mu \in M$. This is a well-ordering relation. For $\lambda \in \Lambda \subset N$, put $C_\lambda = v(B_\lambda)$. Then the family $(C_v)_{v \in N}$ satisfies conditions a), b), c) of Def. 1 with respect to the A-algebra C.