Reflexive Closure is Closure Operator/Proof 2

Theorem
Let $S$ be a set.

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

Then the reflexive closure operator on $R$ is a closure operator.

Reflexive Closure is Inflationary
$\Box$

Reflexive Closure is Order Preserving
$\Box$

Reflexive Closure is Idempotent
$\Box$

Thus by the definition of closure operator, reflexive closure is a closure operator.