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, $g \circ f \left({x}\right) = g \left({f \left({x}\right)}\right) = g \left({y}\right) = z$.

Hence $g \circ f$ is surjective.