Pi-Hall subgroups exist in pi-separable

Statement
Suppose $$\pi$$ is a set of primes and $$G$$ is a finite group that is a $$\pi$$-separable group (see fact about::pi-separable group). Then the following are true:


 * $$G$$ has a $$\pi$$-Hall subgroup.
 * $$G$$ has a $$\sigma$$-Hall subgroup for $$\sigma = \pi \cup \{ q \}$$ where $$q$$ is a prime outside $$\pi$$.
 * $$G$$ has a $$\sigma$$-Hall subgroup for $$\sigma = \{ p,q \}$$ where $$p \in \pi$$ and $$q \notin \pi$$.

Related facts

 * Sylow subgroups exist (part of Sylow's theorem)
 * Hall subgroups exist in finite solvable (part of ECD condition for pi-subgroups in solvable groups)