Proposition 11

Suppose that $p \neq 0$. Let $x, y$ be two permutable elements of $G_f$. Then $xy \in G_f$ and $\log(xy) = \log x + \log y$.

The fact that $xy \in G_f$ follows from Lemma 1. Let $U$ be an open subgroup of the additive group $L(G)$ and $\phi : U \to \phi(U)$ an exponential mapping of $G$ with the properties of Proposition 3 of no. 2; $U$ can be assumed to be small enough for $\log|\psi(U)|$ to be the inverse mapping of $\phi$. For $n \in \mathbf{Z} - \{0\}$ suitably chosen, $x^n \in \phi(U)$, $y^n \in \phi(U)$. Let $u = \log x^n$, $v = \log y^n$, whence $x^n = \phi(u)$, $y^n = \phi(v)$. By formula (2), $[u, v] = 0$. The Hausdorff formula proves then that $\phi(\lambda(u + v)) = \phi(\lambda u) \phi(\lambda v)$ for $|\lambda|$ sufficiently small; hence, for every sufficiently large integer $i$,

that is or