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Welsh's same-paper deductions use the following earlier results. Their use is explicit on printed pp. 1324–1329 (published PDF).

Finite independent representatives

The external theorem attributed to R. Rado, A theorem on independence relations, Quarterly Journal of Mathematics 13 (1942), 83–89, is:

Let $(S,M)$ be a finite matroid with rank function $r$, and let $(C_\ell)_{\ell\in L}$ be a finite indexed family of subsets of $S$. There is an injective assignment $c_\ell\in C_\ell$ whose range is independent in $M$ if and only if

$$r\left(\bigcup_{\ell\in K}C_\ell\right)\ge |K| \qquad(K\subseteq L).$$

For $L=\varnothing$, the empty assignment is the conclusion. This exact finite interface is also recorded in the Rado (1949) external-input page. The 1942 primary paper and its proof have not been acquired or reconstructed in either source unit. Every Welsh result that uses this theorem is therefore a complete relative proof, not a new proof of Rado's theorem.

Transversal and Hall inputs

The statement that partial transversals of a finite indexed family form a matroid has a complete alternating-path proof in Edmonds–Fulkerson's transversal-matroid theorem. Welsh attributes this result independently to Mirsky–Perfect and Edmonds–Fulkerson.

The exact ordinary union criterion used for replicated families is Hall's theorem. The labeled-copy form for a uniform multiplicity bound is also expanded in the Hall replication argument.

Finite independent-set augmentation and extension to a base are proved in elementary finite matroid facts. Together with the transversal-matroid theorem, they justify the local partial-to-full transversal augmentation.

Perfect and common-transversal comparisons

The source states H. Perfect's rank-defect corollary without proof. A complete deduction from the exact finite Rado input is supplied on the local Perfect-corollary page. This does not compile Perfect's unpublished 1967 seminar report.

Welsh cites Ford and Fulkerson's 1962 book for a common-transversal result. The related canonical common-SDR theorem proves the $p_i=q_i=1$ comparison from their 1958 paper. It is not presented as an inspection of the separately cited book.

All remaining partition-matroid, copied-ground-set, rank, and defect calculations used by this unit are proved on the local result pages.