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Source. Rado (1949), Lemma 2, statement on printed p. 340, proof on pp. 340–341 (canonical PDF).

Statement. Let $r$ be a finite-rank function on $M$ satisfying (R1)–(R3), let $I$ be an arbitrary set, and let $A_i\subseteq M$ be finite for every $i\in I$. There are pairwise distinct representatives $a_i\in A_i$ whose whole image is independent if and only if

$$r\left(\bigcup_{i\in F}A_i\right)\ge |F| \quad\text{for every finite }F\subseteq I. \tag{1}$$

Lemma 2 itself asserts the sufficiency of (1); the necessity is the remark printed directly after it ("Clearly, (7) is necessary", p. 340), where (7) is the paper's label for (1). Finiteness of $A_i$ is inherited from the source's notation and is essential. Condition (1) applied to a singleton also ensures that $A_i$ is nonempty. No rank evaluation on an infinite union is assumed.

External input. The exact finite independent-representative theorem is Rado (1942), Theorem 3, as invoked by the 1949 source.

Proof. Necessity follows because, for finite $F$, the independent set of distinct representatives $\{a_i:i\in F\}$ has rank $|F|$ and is contained in $\bigcup_{i\in F}A_i$. Monotonicity of finite rank gives (1).

Conversely, assume (1). For each finite $N\subseteq I$, the family $(A_i)_{i\in N}$ satisfies the finite theorem's hypothesis for every subfamily. That external theorem supplies an injective selection $x_N:N\to M$ with $x_N(i)\in A_i$ and independent image. Choose one such $x_N$ for every finite $N$; for $N=\varnothing$ use the empty map.

Apply Lemma 1 to these local choices. It gives $a_i\in A_i$ such that, for every finite $F\subseteq I$, there is finite $N\supseteq F$ with $a_i=x_N(i)$ for all $i\in F$.

Since $x_N$ is injective, its restriction to $F$ is injective. Since its image is independent, the image of that restriction is independent by heredity. Thus

$$r(\{a_i:i\in F\})=|\{a_i:i\in F\}|=|F|. \tag{2}$$

Taking $F$ to be any two distinct indices proves global injectivity. If $T$ is a finite subset of the global image, its preimage under the injective selection is a finite set $F$, and (2) gives $r(T)=|T|$. This is precisely independence of the whole image. $\square$

Source comparison. The source proves (2) by the equivalent rank sandwich on the two parts of $x_N(N)$. Both subadditivity and hereditary independence are expanded in the linked finite-rank page. The local-to-global step, the distinctness requirement, and the finite-support definition of independence are all retained. This is a complete relative proof; the finite theorem from the separate 1942 paper remains external.