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:

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