Order independence of disjoint analytic plumbing

Form the disjoint union of the original complement with one local fixture for each pair. For pair $j$, let $R_j$ be the relation identifying its two fixture boundary circles with the corresponding boundary circles in the original complement, using the declared $z_j$ and $w_j$ coordinates. Each $R_j$ is supported only on the $j$th pair's coordinate neighborhoods and fixture. Pairwise disjointness means that no point in the support of $R_j$ lies in the support of $R_k$ for $j\mathrm{\ne }k$. Thus quotienting by any initial segment of the relations leaves every remaining boundary identification unchanged, and every successive order produces the quotient by the same union of seam relations. The resulting point sets and quotient maps are canonically identified by the identity on each labeled piece.

On the complement and every fixture, the charts are the same in each order. At a nonzero seam, the only new coordinate transition is $w_j=q_j\mathrm{/}{z}_j$ or its inverse, holomorphic because both coordinates are nonzero there; [F2] supplies the smooth-annulus and nodal local models at every parameter. No transition compares distinct plumbing neighborhoods. Therefore the canonical identity between the quotients preserves every chart and is biholomorphic, and it fixes the parameters, remaining marks and their local coordinates.