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Source. Hall (1935), Theorem 2 and its proof, printed p. 29 (canonical PDF).

Statement. Let $\mathcal P$ be a partition of a set $S$, and let $(T_i)_{i\in[m]}$ be a finite indexed family of subsets of $S$, with $m\ge0$. There are points $a_i\in T_i$ belonging to pairwise distinct classes of $\mathcal P$ if and only if, for every $I\subseteq[m]$,

$$\left|\left\{P\in\mathcal P: P\cap\bigcup_{i\in I}T_i\ne\varnothing\right\}\right| \ge |I|. \tag{1}$$

There is no finiteness assumption on $S$, on the $T_i$, or on $\mathcal P$. The represented family has only $m$ indices. Repeated values among the $T_i$ are permitted. The resulting points themselves are distinct because their partition classes are disjoint.

Proof. Necessity follows because the points with indices in $I$ belong to $|I|$ distinct classes, each meeting $\bigcup_{i\in I}T_i$.

For sufficiency define, for each $i\in[m]$, a subset of the set $\mathcal P$ by

$$\mathcal T_i=\{P\in\mathcal P:P\cap T_i\ne\varnothing\}.$$

For every $I\subseteq[m]$, the union $\bigcup_{i\in I}\mathcal T_i$ is exactly the set of classes in (1): a class meets a union precisely when it meets at least one member. Thus (1) is Hall's union condition for the finite indexed family $(\mathcal T_i)_{i\in[m]}$. By Theorem 1, there are pairwise distinct classes $P_i\in\mathcal T_i$. For each of the finitely many indices define

$$M_i=P_i\cap T_i\ne\varnothing.$$

Choose $a_i\in M_i$ successively for $i=1,\ldots,m$. This requires only finite selection from nonempty sets. Then $a_i\in T_i$, and the distinct selected classes $P_i$ are pairwise disjoint, so the points lie in different classes and are themselves pairwise distinct. When $m=0$ all assignments and choices are empty and the conclusion holds. $\square$

Source precision. The source applies Theorem 1 to its class sets $t_i$ and then names the selected classes $S_1,\ldots,S_m$ after relabeling. Its displayed intersection is written $S_i\cap T_i=M$, but the following sentence calls it $M_i$. The consistent indexed notation above repairs this mismatch. It does not change a hypothesis or conclusion and is not attributed to an author-issued erratum. The numbered source statement gives sufficiency; the reverse direction written here is the immediate necessary condition.

Used by. Theorem 3. The definitions and finite-index qualifications are recorded on the definitions page.

Bears on. No problem. The Problem 126 argument cites only Theorem 1, not this partition form.