= Product-sum comparison lemma
{title2=$\kappa\lambda\leq\kappa+\lambda$}
In ZF, let $K,L$ be nonempty. If $K\times L$ injects into $K\sqcup L$, then there is either an injection or a surjection from $K$ to $L$.
Indeed, extend the inverse of the given injection to a surjection $p:K\sqcup L\to K\times L$ by sending points outside its range to one fixed pair $(k_*,l_*)$. If the second coordinate of $p$ restricted to the $K$-summand covers $L$, it is the required surjection. Otherwise choose $l_0$ that it misses. Every $(k,l_0)$ then has its preimage in the $L$-summand. Except possibly for the default pair, that preimage is unique, so these preimages inject $K$ into $L$. In the exceptional case they inject $K\setminus\{k_*\}$ into $L$; either their image is all of $L$, yielding a surjection from $K$, or an omitted point extends the map to an injection from $K$.
Back to article page
