= Solution
<König theorem for cardinal numbers> states that if $\kappa_i<\lambda_i$ for each index $i\in I$, then
$$
\boxed{\sum_{i\in I}\kappa_i<\prod_{i\in I}\lambda_i.}
$$
Represent each <cardinal> by its <initial ordinal>, let $A=\{(i,\alpha):i\in I,\alpha<\kappa_i\}$, and let $B=\prod_{i\in I}\lambda_i$.
First there is an <injection> $A\to B$: encode $(i,\alpha)$ by the <function> which is zero away from $i$ and takes value $\alpha+1$ at $i$. The nonzero coordinate identifies $i$, and its value identifies $\alpha$. It is valid because $\alpha+1\le\kappa_i<\lambda_i$; an index with $\kappa_i=0$ contributes no domain points.
For strictness suppose a surjection $F:A\to B$ existed. At coordinate $i$, the values $F(i,\alpha)(i)$ for $\alpha<\kappa_i$ occupy fewer than $\lambda_i$ possibilities. Choose $b_i$ as the least <ordinal> below $\lambda_i$ absent from that <set>. The resulting $b=(b_i)\in B$ differs from $F(i,\alpha)$ at coordinate $i$ for every $(i,\alpha)\in A$, contradicting <surjectivity>.
An <injection> $B\to A$ would produce a surjection $A\to B$ by inverting on its image and assigning the zero <function> elsewhere. It is therefore impossible. The <injection> in the first direction and impossibility of the reverse give the strict <cardinal> inequality. This diagonal argument uses the canonical well-orders of the specified <cardinals> to choose missing coordinate values, rather than an unexplained arbitrary choice at that step.
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