Reflexive Closure is Inflationary

Theorem
Let $S$ be a set.

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

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

Proof
Let $\RR \in R$.

The reflexive closure $\RR^=$ of $\RR$ is defined as:


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

From Set is Subset of Union:


 * $\RR \subseteq \RR^=$

Hence the reflexive closure operator is an inflationary mapping.