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We have the following indirect implication of form equivalence classes:

1 \(\Rightarrow\) 406 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
1 \(\Rightarrow\) 406

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
1:	\(C(\infty,\infty)\):  The Axiom of Choice: Every  set  of  non-empty sets has a choice function.
406:	The product of compact Hausdorf spaces is countably compact. Alas [1994].

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