Union with Relative Complement

Theorem
The union of a set $$T$$and its relative complement in $$S$$ is the set $$S$$:

$$T \cup \mathcal{C}_S \left({T}\right) = S$$

Proof
$$ $$

From the definition of relative complement, we have that $$T \subseteq S$$.

From Subset Equivalences, we have that $$T \subseteq S \iff S \cup T = S$$, from which the result follows.