Proposition 4.1

For every profinite group $G$ in universe $u$ and closed subgroups $N,H,K\le G$, suppose that $N$ is normal in $G$ and its multiplication is commutative, and that every element of $G$ can be expressed both as $nh$ with $n\in N,h\in H$ and as $n'k$ with $n'\in N,k\in K$. Suppose also that for every natural prime $p$ there exist a Sylow pro-$p$ subgroup $P_p$ of $K$ and an element $g_p\in G$ with $g_pP_pg_p^{-1}\le H$. Then there exists a single $g\in G$ with $gKg^{-1}\le H$. A Sylow pro-$p$ subgroup $P$ of a subgroup $A\le G$ means a subgroup $P\le A$ that is closed in $G$, for which every quotient $P/U$ by an open normal subgroup of $P$ has the property that every element is killed by some power $p^k$ with $k\in\mathbb N$, and that is maximal under inclusion among the closed subgroups of $G$ contained in $A$ with this quotient property. The product expressions need not be unique, and neither intersection with $N$ is required to be trivial. The conclusion is containment rather than equality. Neither the local subgroup nor its conjugating element must be independent of $p$. All natural primes are included; trivial groups, trivial $N$, and trivial Sylow subgroups are permitted, and none of these groups is required to be finite.