# Right Inverse Mapping is Injection

## Theorem

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

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

Then $g$ is an injection.

## Proof

By the definition of right inverse:

$f \circ g = I_T$

where $I_T$ is the identity mapping on $T$.

By Identity Mapping is Injection, $I_T$ is an injection.

By Injection if Composite is Injection, it follows that $g$ is an injection.

$\blacksquare$