I do not understand what is going on in $(4)$: for every flat $E \in \mathcal F$, $E \ne X$, the flats that cover $E$ in $\mathcal F$ partition the remaining parts.
What is meant by "the flats that cover $E$ in $\mathcal F$ partition the remaining parts."?
I will try to explain this via an analogy to (linear) geometry into two dimensions, call the space $L$.
You will have 4 "types" of flats: the empty set $\phi$, single points $\{\{p\}: p \in L\}$ and all lines $\{\{\ell\}: \ell \in L\}$, and $L$ itself. Single points cover the empty flat, a line that contains a point covers that point, and L covers each line.
Lets fix a point, call it $p$. Then any two lines $\ell_1$ and $\ell_2$ that cover $p$ (i.e., contain $p$ as a point) are disjoint when $p$ is excluded. Moreover, every point in $L-\{p\}$ will be contained in some line that covers $p$ (i.e., some line through $p$). Thus all the lines through $p$ (i.e., all the flats that cover $p$) partition the remaining points (i.e., the lines through $p$ contain all points in $L-\{p\}$ and no two lines intersect in $L-\{p\}$).
Try to relate this "inclusion" structure for families of subsets when working with discrete geometries.