We have the following indirect implication of form equivalence classes:

95-F \(\Rightarrow\) 173
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
95-F \(\Rightarrow\) 67 Some theorems on vector spaces and the axiom of choice, Bleicher, M. 1964, Fund. Math.
The Axiom of Choice, Jech, 1973b, page 148 problem 10.4
67 \(\Rightarrow\) 76 clear
76 \(\Rightarrow\) 173 Paracompactness of metric spaces and the axiom of choice, Howard, P. 2000a, Math. Logic Quart.

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

Howard-Rubin Number Statement
95-F:

Existence of Complementary Subspaces over a Field \(F\): If \(F\) is a field, then every vector space \(V\) over \(F\) has the property that if \(S\subseteq V\) is a subspace of \(V\), then there is a subspace \(S'\subseteq V\) such that \(S\cap S'= \{0\}\) and \(S\cup S'\) generates \(V\). H. Rubin/J. Rubin [1985], pp 119ff, and Jech [1973b], p 148 prob 10.4.

67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

76:

\(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).

173:

\(MPL\): Metric spaces are para-Lindelöf.

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