Alexander's theorem: every link is a closed braid

Choosing a diagram and a first reduction. If $L$ is empty, its empty diagram is read by [F4] as the empty braid in $B_0$, proving the assertion. For a nonempty $L$, by [F1] fix a regular projection $D_0$ of $L$ with its over/under and orientation data, and let $h_0:=h(D_0)\in \mathbb{N}$. If $h_0=0$, [F4] already presents $L$ as the closure of a braid. If $h_0>0$, then by [F2] the Seifert picture of $D_0$ contains a defect region and a reducing arc; performing the reducing move produces a diagram $D_1$ of the same oriented link with $h(D_1)=h_0-1$ by [F3].

Termination of the algorithm. Iterate step 1.1. The sequence of heights is a strictly decreasing sequence of nonnegative integers, because each reducing move is a Reidemeister II move of the diagram, which does not change the represented oriented link, and lowers the height by exactly one; hence after exactly $h_0$ steps the algorithm stops at a diagram $D_{h_0}$ with $h(D_{h_0})=0$ representing $L$.

Reading the braid. By [F4] the height-zero diagram $D_{h_0}$ is put in closed-braid form by a sphere isotopy and a choice of planar chart; the braid word read from the nested chain has a closure equivalent to the link of $D_{h_0}$, which is $L$. Hence $L$ is equivalent to the oriented closure $\widehat{\beta }$ of an explicit geometric braid $\beta$ on $n\ge 1$ strands.

Conclusion. Steps 1.1-3.1 give the required braid. AC is used in [F1] (regular projections, through the bridge from AC to ${\mathrm{A}\mathrm{C}}_{\omega }$) and in the AC-stated height and reducing-move chain [F2], [F3], which rests on the annulus lemma.