= Solution
<König theorem for cardinal numbers> says that for families of <cardinals> $\kappa_i<\lambda_i$ indexed by a set $I$,
$$
\boxed{\sum_{i\in I}\kappa_i<\prod_{i\in I}\lambda_i.}
$$
The sum is the <cardinality> of the disjoint union of sets $A_i$ of sizes $\kappa_i$, and the product is the <cardinality> of $\prod_iB_i$ for sets of sizes $\lambda_i$.
Consider any function $F:\bigsqcup_iA_i\to\prod_iB_i$. For each $i$, the set of $i$th coordinates of images of <points> from the $i$th block has <cardinality> at most $\kappa_i<\lambda_i$. Choose $b_i\in B_i$ outside it. Then $b=(b_i)_i$ is not the image of any <point>: a <point> from block $i$ has the wrong $i$th coordinate. Thus there is no <surjection> from the sum set onto the product set. <Cardinal> comparability in $ZFC$ makes the inequality strict in the indicated direction. For the empty index set the statement is $0<1$, also true. This is the diagonal argument underlying the inequality.
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