  A K\"onig-Egerv\'ary graph is a graph $G$ satisfying
$\alpha(G)+\mu(G)=|V(G)|$, where $\alpha(G)$ is the cardinality of a maximum
independent set and $\mu(G)$ is the matching number of $G$. Such graphs are
those that admit a matching between $V(G)-\bigcup \Gamma$ and $\bigcap \Gamma$
where $\Gamma$ is a set-system comprised of maximum independent sets satisfying
$|\bigcup \Gamma'|+|\bigcap \Gamma'|=2\alpha(G)$ for every set-system $\Gamma'
\subseteq \Gamma$; in order to improve this characterization of a
K\"onig-Egerv\'ary graph, we characterize \emph{hereditary K\"onig-Egerv\'ary
set-systems} (HKE set-systems, here after).
  An \emph{HKE} set-system is a set-system, $F$, such that for some positive
integer, $\alpha$, the equality $|\bigcup \Gamma|+|\bigcap \Gamma|=2\alpha$
holds for every non-empty subset, $\Gamma$, of $F$.
  We prove the following theorem: Let $F$ be a set-system. $F$ is an HKE
set-system if and only if the equality $|\bigcap \Gamma_1-\bigcup
\Gamma_2|=|\bigcap \Gamma_2-\bigcup \Gamma_1|$ holds for every two non-empty
disjoint subsets, $\Gamma_1,\Gamma_2$ of $F$.
  This theorem is applied in \cite{hke},\cite{broken}.
