Surjection if Composite is Surjection

Theorem
Let $$f: S_1 \to S_2$$ and $$g: S_2 \to S_3$$ be mappings such that $$g \circ f$$ is a surjection.

Then $$g$$ is a surjection.

Proof
Let $$g \circ f$$ be surjective.

Then $$\forall z \in S_3: \exists x \in S_1: g \circ f \left({x}\right) = z$$.

Then:

$$ $$ $$ $$

Also see

 * Injection if Composite is an Injection