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Source. Equation (14) and its defect-Hall explanation in the proof of Theorem 12, printed pp. 1328–1329 (published PDF).

Statement. Let $T_p(\mathcal A)$ be the transversal matroid of the replicated family $\mathcal A^p$, whether or not a full $p$-transversal exists. For every $X\subseteq S$,

$$r_p(X) =\min_{J\subseteq I} \bigl(|X\cap A(J)|+N-p(J)\bigr), \tag{1}$$

where $N=p(I)$. Equivalently, for every integer $t$,

$$r_p(X)\ge t \quad\Longleftrightarrow\quad |X\cap A(J)|\ge p(J)+t-N \quad(J\subseteq I). \tag{2}$$

If $\mathcal A$ has a $p$-transversal, then $r_p(S)=N$ and the bases of $T_p(\mathcal A)$ are exactly the $p$-transversals.

Proof. The rank formula (1) is Welsh's equation (14), which the source obtains from the defect version of Hall's theorem applied to the replicated family (pp. 1328–1329). The equivalence (2) restates (1) threshold by threshold.

The final base assertion is the corrected Theorem 3. $\square$