Set Theory/Examples/A cup (X cap B) = C, (A cup X) cap B = D/Converse

Example in Set Theory
Let $A, B, C, D$ be subsets of a set $S$.

Let the following conditions hold:
 * $A \cap B \subseteq D \subseteq B$

and:
 * $A \cup D = C$

Then these equations hold:
 * $A \cup \paren {X \cap B} = C$
 * $\paren {A \cup X} \cap B = D$

if:
 * $X = D \setminus A$

Proof
Suppose $X = D \setminus A$.

Then: