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Source. Rado (1949), the theorem in §4, part (ii), statement on printed p. 341 and proof on pp. 342–343 (canonical PDF).

Statement. Part (ii) of the Theorem: If $L_1\subseteq L\subseteq M$ and $L_1$ is independent, there is a base $B$ of $L$ containing $L_1$. In particular, every subset of $M$, including the empty set, has a base.

Proof. Let

$$\mathcal P=\{J:L_1\subseteq J\subseteq L,\ J\text{ independent}\},$$

partially ordered by inclusion. It is a set and is nonempty because $L_1\in\mathcal P$.

Let $\mathcal C\subseteq\mathcal P$ be a nonempty chain and set $J_*=\bigcup_{J\in\mathcal C}J$. Then $L_1\subseteq J_*\subseteq L$. Every nonempty finite $F\subseteq J_*$ is contained in one member of $\mathcal C$: first choose, for each element of $F$, a chain member containing it, then take the largest of these finitely many members under inclusion. That member is independent, so $r(F)=|F|$. For $F=\varnothing$ the same equality is (R1). Thus $J_*$ is independent and lies in $\mathcal P$, where it is an upper bound for $\mathcal C$. The empty chain has the upper bound $L_1\in\mathcal P$.

The exact Zorn lemma therefore supplies a maximal $B\in\mathcal P$. If an independent set $B'$ satisfied $B\subsetneq B'\subseteq L$, it would still contain $L_1$, so it would contradict maximality in $\mathcal P$. Hence $B$ is a base of $L$. Taking $L_1=\varnothing$ gives the last assertion. $\square$

Source correction. On printed p. 343 the chain is denoted $\Lambda'$ and its union $L^{\prime\prime\prime}$. The source prints $L^{\prime\prime\prime}\in\Lambda'$. What has been proved, and what Zorn's lemma requires, is $L^{\prime\prime\prime}\in\Lambda$, the ambient poset. A chain need not contain its own union. For example, the increasing chain of finite initial segments of $\mathbb N$ has union $\mathbb N$, which is not one of those segments. The proof above uses the correct ambient membership and also treats the empty chain. This is a compilation clarification, not a claim of an author-issued erratum.

This part uses only finite character of independence and Zorn's lemma. It does not depend on the representative theorem or on augmentation.