Corollary 1.3

For every profinite group $G$ in universe $u$ and closed subgroups $N,J,H\le G$, assume that $N$ is normal in $G$, that every element of $G$ has a unique expression $nj$ with $n\in N$ and $j\in J$, that $N$ is pronilpotent, and that either $G$ is prosupersolvable or $J$ is pronilpotent. Assume also that $N\cap H$, regarded as a subgroup of $N$, is normal in $N$, and that for every natural prime $p$ there exist a Sylow pro-$p$ subgroup $P_p$ of $J$ and an element $g_p\in G$ with $g_pP_pg_p^{-1}\le H$. Then there exists one $g\in G$ such that $gJg^{-1}\le H$. 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. Saying that a topological group $R$ is pronilpotent means that $R/U$ is nilpotent for every open normal subgroup $U$ of $R$, with the subgroup and quotient topologies understood. For a group $R$, the series condition used here means that there exist $m\in\mathbb N$ and a nondecreasing sequence $(S_i)_{i\in\mathbb N}$ of normal subgroups of $R$ such that $S_0=\{1\}$, $S_m=R$, and, for each $i<m$, some $r_i\in R$ satisfies $S_{i+1}=\langle S_i,r_i\rangle$. Repeated terms are allowed, and $m=0$ is allowed precisely when $R$ is trivial. Saying that $R$ is prosupersolvable means that every quotient $R/U$ by an open normal subgroup satisfies this series condition. The normality assumption on $N\cap H$ is in $N$, not a requirement of normality in $G$. The local witnesses can vary with $p$, whereas the conclusion concerns one conjugate of the whole subgroup $J$. All primes are quantified, including those with trivial Sylow subgroups; trivial groups and trivial factors in the unique product decomposition are permitted.