On the existence of minimal $\tau$-closed totally saturated non-$\mathfrak H$-formations

The article deals with finite groups. A $\tau$-closed totally saturated formation $\mathfrak F$ is called a minimal $\tau$-closed totally saturated non-$\mathfrak H$-formation (or an $\mathfrak H_\infty^\tau$-critical formation) if $\mathfrak F\not\subseteq\mathfrak H$, but all proper $\tau$-closed totally saturated subformations of $\mathfrak F$ are contained in $\mathfrak H$. Theorem. Let $\mathfrak F$ and $\mathfrak H$ be $\tau$-closed totally saturated formations, $\mathfrak F\not\subseteq\mathfrak H.$ Then $\mathfrak F$ has at least one minimal $\tau$-closed totally saturated non-$\mathfrak H$-formation.