Uniqueness in Weierstrass preparation

By [L1], after shrinking to a common neighbourhood the unit factors $u$ and $v$ are nowhere zero. Therefore for each fixed nearby parameter $z^{\mathrm{\prime }}$, the slice zeros of $f(z^{\mathrm{\prime }},\cdot )$ coincide, with multiplicity, with the slice zeros of $W(z^{\mathrm{\prime }},\cdot )$ and also with those of $P(z^{\mathrm{\prime }},\cdot )$.

Fix such a parameter $z^{\mathrm{\prime }}$. By [L2], both $W(z^{\mathrm{\prime }},\cdot )$ and $P(z^{\mathrm{\prime }},\cdot )$ are monic degree-$d$ one-variable polynomials with the same multiset of roots, counted with multiplicity. Over $\mathbb{C}$, a monic polynomial is the product of its linear factors, so these two polynomials are equal. Since this holds for every nearby $z^{\mathrm{\prime }}$, the germs satisfy $W=P$.

With $W=P$, the two preparations give $(u-v)W=0$. On the nonempty open set where $W\mathrm{\ne }0$ one therefore has $u=v$. Applying [L3] to the holomorphic function $u-v$ on the connected neighbourhood shows $u=v$ everywhere there, and hence as germs.