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Source. Rado (1949), the theorem in §4, part (iii), and the paragraph defining rank cardinal, printed p. 341; conclusion of the proof on p. 343 (canonical PDF).

Statement. Part (iii) of the Theorem: any two bases of a set $L\subseteq M$ have equal cardinality. The paragraph after the Theorem (p. 341) adds, using parts (ii) and (iii), that this common cardinal is the largest cardinality of an independent subset of $L$, defines it as the rank cardinal $r(L)$, and notes that for finite $L$ it agrees with the original finite-rank function.

Proof. Let $B_1,B_2$ be bases of $L$. If their cardinalities differed, cardinal comparability in the stipulated choice setting would allow us to relabel them so that $|B_1|<|B_2|$. By part (i), some $x\in B_2\setminus B_1$ makes $B_1\cup\{x\}$ independent. It is a larger independent subset of $L$, contradicting that $B_1$ is a base. Therefore $|B_1|=|B_2|$.

By part (ii), a base $B$ exists. Every independent $J\subseteq L$ extends to some base $B_J$ of $L$, so

$$|J|\le |B_J|=|B|.$$

The upper bound is attained by the independent set $B$ itself. Thus this is a largest cardinal, not merely a supremum that might fail to be attained. Its definition is independent of the chosen base.

If $L$ is finite, the finite-base calculation gives $r(L)=|B|$ for the original integer-valued function. Hence the new cardinal-valued definition extends it consistently. For $L=\varnothing$, the unique base is empty and the value is zero. $\square$

This is the paper's conclusion for arbitrary set cardinalities. It does not assert that bases are unique as subsets, or that one can replace finite-character independence by an arbitrary infinite dependence notion. The proof remains relative to the exact external finite selection input and choice assumptions recorded in this unit.