Axiom Of Choice

We have the following indirect implication of form equivalence classes:

391 \(\Rightarrow\) 323 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
391 \(\Rightarrow\) 399	clear
399 \(\Rightarrow\) 323	clear

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

Howard-Rubin Number	Statement
391:	\(C(\infty,LO)\): Every set of non-empty linearly orderable sets has a choice function.
399:	\(KW(\infty,LO)\), The Kinna-Wagner Selection Principle for a set of linearly orderable sets: For every set of linearly orderable sets \(M\) there is a function \(f\) such that for all \(A\in M\), if \(\|A\|>1\) then \(\emptyset\neq f(A)\subsetneq A\).
323:	\(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set \(M\) of well orderable sets there is a function \(f\) such that for all \(A\in M\), if \(\|A\| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

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