Reflexive Closure is Idempotent

Theorem
Let $S$ be a set.

Let $R$ denote the set of all endorelations on $S$.

Then the reflexive closure operator is an idempotent mapping on $R$.

That is:


 * $\forall \RR \in R: \RR^= = \paren {\RR^=}^=$

where $\RR^=$ denotes the reflexive closure of $\RR$.

Proof
Let $\RR \in R$.

By the definition of reflexive closure:


 * $\RR^= = \RR \cup \Delta_S$


 * $\paren {\RR^=}^= = \paren {\RR \cup \Delta_S} \cup \Delta_S$

By Union is Associative:


 * $\paren {\RR^=}^= = \RR \cup \paren {\Delta_S \cup \Delta_S}$

By Union is Idempotent:


 * $\paren {\RR^=}^= = \RR \cup \Delta_S$

Hence:


 * $\forall \RR \in R: \RR^= = \paren {\RR^=}^=$