The supports of a multiple test blow-up stay inside the strict transforms of a hypersurface of maximal contact

Base case. By [F1], each point of $\mathrm{supp}(\mathcal{I},\mu )$ lies in $\mathrm{supp}({\mathcal{D}}^{\mu -1}(\mathcal{I}),1)$, so the section $u\in {\mathcal{D}}^{\mu -1}(\mathcal{I})(U)$ vanishes there. Hence $\mathrm{supp}(\mathcal{I},\mu )\subseteq V(u)$, which is the case $i=0$.

Inductive step. Assume $\mathrm{supp}({\mathcal{I}}_i,\mu )\subseteq V(u_i)$. The next center satisfies $C_i\subseteq \mathrm{supp}({\mathcal{I}}_i,\mu )\subseteq V(u_i)$. If $C_i$ is Cartier, work locally near it with a regular centre equation $y$. Since $C_i$ is reduced and $u_i$ vanishes on it, $u_i=ay$; multiplicity one of $u_i$ along the centre makes $a$ a unit there. Although the blow-up is an isomorphism, the controlled transform divides by $y$: [F5] gives $u_i\mathrm{/}y=a\in {\mathcal{D}}^{\mu -1}({\mathcal{I}}_{i+1})$. Thus this derivative ideal is the unit ideal locally along the centre, and [F1] makes the new support empty. The strict transform of $V(u_i)$ is also empty there because $V(u_i)=C_i$ locally. Away from the centre, both transforms preserve the prior containment. If $C_i$ is not Cartier and is a proper subset of the support, [F2] gives a tangent direction $u_{i+1}$ whose zero scheme is the strict transform $V(u{)}_{i+1}$. In the remaining case, work componentwise where $C_i$ has codimension at least two. Since $u_i$ vanishes on $C_i$ and has order one, it is one of local regular parameters generating the ideal of $C_i$. On the chart indexed by $u_i$, the controlled transform $u_{i+1}=u_i\mathrm{/}y$ is $1$, and [F4] says the strict transform of $V(u_i)$ is empty there. By [F5] with $r=\mu -1$, the controlled transform of $({\mathcal{D}}^{\mu -1}({\mathcal{I}}_i),1)$ is contained in ${\mathcal{D}}^{\mu -1}({\mathcal{I}}_{i+1})$. Since $u_i\in {\mathcal{D}}^{\mu -1}({\mathcal{I}}_i)$, its controlled transform $u_{i+1}=y^{-1}{\sigma }_i^{\ast }(u_i)$ lies in that derivative ideal; [F5] applies because $C_i\subseteq \mathrm{supp}({\mathcal{I}}_i,\mu )$, including when equality holds. On the chart indexed by $u_i$, $y=u_i$, so $u_{i+1}=1$; the derivative ideal is the unit ideal, and [F1] makes the transformed support empty there, as required by the empty strict-transform chart in [F4]. On a chart indexed by another center parameter $t_j$, $y=t_j$ and $u_{i+1}=u_i\mathrm{/}{t}_j$ is a chart coordinate; [F5] places it in ${\mathcal{D}}^{\mu -1}({\mathcal{I}}_{i+1})$, so [F1] puts the support in $V(u_{i+1})$, the strict transform by [F4]. This proves the containment for this center as well. Applying [F1] at stage $i+1$ closes the induction for all stages. For the restricted-sequence conclusion, assume $u$ transversal to the initial boundary. The distinct equations $u$ and the boundary parameters then form part of one regular parameter system. Since each regular center lies in $V(u_i)$ and has SNC with the boundary, its ideal can be generated by $u_i$ and further parameters compatible with the boundary restrictions. The nonempty charts in [F4] restricted to $V(u_i)$ are precisely the charts of its center blowup, the restricted exceptional equation is the same $y$, and division by $y^{\mu }$ commutes with restriction. These charts preserve SNC of the restricted boundary; where a component is deleted there is no stalk to check. Thus the restricted ideals and boundaries form the asserted multiple test blow-up.