  We show that for every finitely generated closed subgroup $K$ of a
non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$
containing $K$, and a continuous homomorphism $\tau \colon U \to K$ satisfying
$\tau(k) = k$ for every $k \in K$. We prove that the intersection of a pair of
finitely generated closed subgroups of a Demushkin group is finitely generated
(giving an explicit bound on the number of generators). Furthermore, we show
that these properties of Demushkin groups are preserved under free pro-$p$
products, and deduce that Howson's theorem holds for the Sylow subgroups of the
absolute Galois group of a number field. Finally, we confirm two conjectures of
Ribes, thus classifying the finitely generated pro-$p$ M. Hall groups.
