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The same-paper arguments are written in full on their result pages. The following inputs retain the external scope they have in Rado's published 1949 proof.

Finite independent representatives

R. Rado, A theorem on independence relations, Quarterly Journal of Mathematics 13 (1942), 83–89, Theorem 3, is invoked explicitly in footnote 8 on printed p. 341 of the 1949 paper. The exact finite form required, in the 1949 rank notation, is:

Let $A_1,\ldots,A_m$ be finite subsets of a set $M$ with a finite-rank function satisfying (R1)–(R3). If

$$r\left(\bigcup_{j\in J}A_j\right)\ge |J| \qquad(J\subseteq\{1,\ldots,m\}),$$

then there are pairwise distinct $a_j\in A_j$ such that $\{a_1,\ldots,a_m\}$ is independent. For $m=0$ the empty selection satisfies the assertion. This is the sole finite selection theorem imported in Lemma 2.

The 1949 source states that Whitney's finite-rank/independence equivalence puts its finite case within that theorem. The needed elementary rank consequences are proved in finite-rank facts. The exact interface above is verified from Rado's 1949 invocation; the 1942 primary paper and its proof have not been acquired or reconstructed in this source unit. In particular, we do not count the finite independent-representative theorem as a new complete proof here.

Choice, recursion, and maximality

We work in ordinary set theory with the axiom of choice. We use the following standard forms explicitly:

• Every set admits a well-order. This supplies the orders of the index and value sets in Lemma 1.
• Transfinite recursion along a set-sized ordinal defines the successive selected values, and transfinite induction proves the invariant. At a limit stage the proof uses only finitely many earlier coordinates.
• Simultaneous choices from set-indexed nonempty families are allowed. This is used to choose local finite representatives and dependent finite supports. We do not infer those choices from a countability assumption.
• Zorn's lemma: a nonempty partially ordered set in which every chain has an upper bound in the poset has a maximal element. It is the exact external input for base extension. Rado cites M. Zorn, A remark on method in transfinite algebra, Bulletin of the American Mathematical Society 41 (1935), p. 667.
• In this choice setting, cardinals are comparable; an injection from $X$ into $Y$ implies $|X|\le |Y|$. These are the cardinal facts used in augmentation and equal cardinality of bases. An ordinal indexing a recursion is not itself asserted to equal a rank cardinal.

No claim is made that these assumptions are logically minimal. In particular, the 1949 footnote that the well-order of the value set can be avoided is not a claim that the whole proof avoids choice.

Other historical references

Whitney's On the abstract properties of linear dependence, American Journal of Mathematics 57 (1935), 509–533, is the source's reference for the finite rank axioms and their independence formulation. Only the deductions used here have been expanded; no full Whitney source compilation is claimed. Steinitz's field-theoretic work is cited as historical context in the introduction. The 1949 paper gives no separate field-theoretic application proof, and none is credited here.

The de Bruijn–Erdős graph theorem and later Euclidean compactness applications are consequences of this source's Lemma 1, not inputs to its proof. Their existing proof pages are linked from that lemma.