# Composition of Right Inverse with Mapping is Idempotent

## Theorem

Let $f: S \to T$ be a mapping.

Let $g: T \to S$ be a right inverse mapping of $f$.

Then:

$\paren {g \circ f} \circ \paren {g \circ f} = g \circ f$

## Proof

 $\displaystyle \paren {g \circ f} \circ \paren {g \circ f}$ $=$ $\displaystyle g \circ \paren {f \circ g} \circ f$ Composition of Mappings is Associative $\displaystyle$ $=$ $\displaystyle g \circ I_T \circ f$ Definition of Right Inverse Mapping $\displaystyle$ $=$ $\displaystyle g \circ f$ Definition of Identity Mapping

$\blacksquare$