Composite of Surjections is Surjection

Theorem
A composite of surjections is a surjection.

That is:
 * If $g$ and $f$ are surjections, then so is $g \circ f$.

Proof
Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be surjections.

Then:

By definition of a composite mapping:
 * $\map {g \circ f} x = \map g {\map f x} = \map g y = z$

Hence $g \circ f$ is surjective.