# Injection if Composite is Injection

## Theorem

Let $f$ and $g$ be mappings such that their composite $g \circ f$ is an injection.

Then $f$ is an injection.

## Proof

Let $g \circ f$ be injective.

We need to show that $\map f a = \map f b \implies a = b$.

So suppose $\map f a = \map f b$.

Then:

 $\ds \map {g \circ f} a$ $=$ $\ds \map g {\map f a}$ Definition of Composition of Mappings $\ds$ $=$ $\ds \map g {\map f b}$ by hypothesis $\ds$ $=$ $\ds \map {g \circ f} b$ Definition of Composition of Mappings

and so $a = b$ as $g \circ f$ is an injection.

So we have shown that:

$\map f a = \map f b \implies a = b$

Hence the result from the definition of injection.

$\blacksquare$