On factorizations of subformations of one-generated hereditary $\omega$-saturated formations

The product $\mathfrak{MH}$ of formations $\mathfrak M$ and $\mathfrak H$ is the class of groups $(G\mid G^\mathfrak H\in\mathfrak M)$. Let $\mathfrak{MH}\subseteq\mathfrak F$, where $\mathfrak F=s^\omega\mathrm{form}G$ is a one-generated hereditary $\omega$-saturated formation. We prove that $\mathfrak M$ is a soluble formation if formations $\mathfrak M$ and $\mathfrak H$ are such that $\mathfrak H\ne\mathfrak{MH}$.