# Inverse of Inverse Relation

## Theorem

The inverse of an inverse relation is the relation itself:

$\paren {\RR^{-1} }^{-1} = \RR$

## Proof

 $\ds \tuple {s, t}$ $\in$ $\ds \RR$ $\ds \leadstoandfrom \ \$ $\ds \tuple {t, s}$ $\in$ $\ds \RR^{-1}$ Definition of Inverse Relation $\ds \leadstoandfrom \ \$ $\ds \tuple {s, t}$ $\in$ $\ds \paren {\RR^{-1} }^{-1}$ Definition of Inverse Relation

$\blacksquare$