Independence systems

Question

Can someone give me a hint on this please, thanks!

Let $(E, F)$ be an independence system. Set $A ⊆ E$ is called maximal F-independent if $A \in F$ and there is no A' ∈ F with A ⊆ A' and A != A'. Let G consist of all subsets C ⊆ E for which C ∩ A = ∅ holds for some maximal F-independent set A. It is then known that (E, G) is an independence system. Further, let H consist of all subsets D ⊆ E for which B ∩D = ∅ holds for some maximal G-independent set B. Show that F = H.

Answer 1

This is because $E \setminus (E \setminus A) = A$ for any $A \subseteq E$. More precisely:

A maximal subset in $G$ must have the form $E \setminus A$ for some maximal subset $A$ of $F$. Similarly a maximal subset of $H$ is $E \setminus B$ for some maximal subset $B = E\setminus A$ of $G$. So every maximal subset of $H$ is of the form $E \setminus (E \setminus A) = A$ for some maximal subset $A$ of $F$.