Composite of Surjections is Surjection

Theorem
A composite of surjections is a surjection.

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

By definition of a composite mapping, $$f \circ g \left({x}\right) = f \left({g \left({x}\right)}\right) = f \left({y}\right) = z$$.

Hence $$f \circ g$$ is surjective.