Set Difference and Intersection form Partition/Corollary 1

Corollary to Set Difference and Intersection form Partition
Let $S$ and $T$ be sets such that:
 * $S \setminus T \ne \varnothing$
 * $T \setminus S \ne \varnothing$
 * $S \cap T \ne \varnothing$

Then $S \setminus T$, $T \setminus S$ and $S \cap T$ form a partition of $S \cup T$, the union of $S$ and $T$.

Proof
From Set Difference and Intersection form Partition:
 * $S \setminus T$ and $S \cap T$ form a partition of $S$
 * $T \setminus S$ and $S \cap T$ form a partition of $T$

From Set Difference Disjoint with Reverse:
 * $\left({S \setminus T}\right) \cap \left({T \setminus S}\right) = \varnothing$

So:
 * $S \cup T = \left({S \setminus T}\right) \cup \left({S \cap T}\right) \cup \left({T \setminus S}\right) \cup \left({S \cap T}\right)$

and the result follows.