Aleph-one-Suslin decomposition into analytic sets (${\aleph }_1$-Suslin)

In ZFC, every ${\aleph }_1$-Suslin set is a union of ${\aleph }_1$ many analytic sets. A branch through a representing tree uses a countable sequence of countable ordinals. That sequence is bounded below ${\omega }_1$, so restrict the labels successively below $\alpha <{\omega }_1$.

Each restricted label set is countable and can be encoded by natural numbers; its projection is therefore an analytic set. The union of these projections is the original set. This is the $\kappa ={\aleph }_0$ case of Successor-Suslin decomposition. The proof explicitly uses the uncountable cofinality of ${\omega }_1$ and countable recoding, hypotheses that need care without the axiom of choice.