Set Difference and Intersection form Partition/Corollary 1

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

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 is Disjoint with Reverse:
 * $\paren {S \setminus T} \cap \paren {T \setminus S} = \O$

So:
 * $S \cup T = \paren {S \setminus T} \cup \paren {S \cap T} \cup \paren {T \setminus S} \cup \paren {S \cap T}$

and the result follows.