Corollary 2

Let G be a topological group and V an open neighbourhood of e. Suppose that V has an analytic manifold structure which makes it into a real (resp. p-adic) Lie group germ. Then G is a real (resp. p-adic) Lie group.

Let $g \in G$. There exists an open neighbourhood $V'$ of $e$ in G such that $V' \cup gV'g^{-1} \subset V$. The mapping $v \mapsto gv g^{-1}$ of $V'$ into $V$ is a continuous and hence analytic morphism of the Lie group germ $V'$ into the Lie group germ $V$. It then suffices to apply Proposition 18 of § 1, no. 9.