On the coincidence of the classes of finite groups $E_{\pi_x}$ and $D_{\pi_x}$

Let $\pi_x$ be the set of primes greater than $x$. We prove that for all $x\in {\Bbb R}$ the classes of finite groups $D_{\pi_x}$ and $E_{\pi_x}$ coincide; i.e., a finite group $G$ possesses a $\pi_x$-Hall subgroup if and only if $G$ satisfies the complete analog of the Sylow Theorems for a $\pi_x$-subgroup.