If $A,B,C$ are sets and $|A \cup B| = |C \times C|$, prove that there is either a surjection from $A$ to $C$, or an injection from $C$ to $B$
Set theory proof, $\exists g(g\colon A\twoheadrightarrow C)$ or $\exists h(h\colon C\hookrightarrow B)$