Relative Complement of Relative Complement/Proof 1

Proof
By the definition of relative complement:
 * $\relcomp S {\relcomp S T} = S \setminus \paren {S \setminus T}$

Let $t \in T$.

Then by the definition of set difference:
 * $t \notin S \setminus T$

Since $T \in T$ and $T \subseteq S$, by the definition of subset:
 * $t \in S$

Thus:
 * $t \in \paren {S \setminus \paren {S \setminus T} }$

Suppose instead that:
 * $t \in \paren {S \setminus \paren {S \setminus T} }$

Then:
 * $t \in S$

and:
 * $\neg \paren {t \in \paren {S \setminus T} }$

Thus:
 * $\neg \paren {\paren {t \in S} \land \neg \paren {t \in T} }$

By Implication Equivalent to Negation of Conjunction with Negative:


 * $t \in S \implies t \in T$

By Modus Ponendo Ponens:


 * $t \in T$