Relative tubular neighborhoods preserve a chain of submersions

Fix $s\in S$ and put $I(s)=\{i:s\in C_i\}$, a finite set. Since the $C_i$ with $i\mathrm{\notin }I(s)$ are closed in $S$ and avoid $s$, and the $U_i$ with $i\in I(s)$ are open, choose a source neighbourhood $V_s\subseteq S$ that avoids every $C_i$ with $i\mathrm{\notin }I(s)$ and lies in every $U_i$ with $i\in I(s)$; shrink it so that the active maps factor smoothly through a single maximal map $F_j$. Since $F_j{\mathrm{\mid }}_S$ is a submersion, there are ambient coordinates $(b,u,z)$ on a neighbourhood of $s$ with $S=\{z=0\}$ and $F_j=b$: straighten $S$, use [F1] on $F_j{\mathrm{\mid }}_S$ for coordinates $(b,u)$, and then apply [F1] in the full-rank case to the map $(b,u,z)\mapsto (F_j(b,u,z),u,z)$, whose differential at $s$ is invertible because $\mathrm{\partial }{F}_j\mathrm{/}\mathrm{\partial }b$ is the identity at $z=0$. If $I(s)$ is empty, ordinary straightened coordinates suffice.

Put $Q=T{\mathbb{R}}^N{\mathrm{\mid }}_S\mathrm{/}TS$ and identify it with the Euclidean orthogonal normal bundle. The $z$-directions of the adapted charts give smooth local splittings $j_s:Q\to T{\mathbb{R}}^N{\mathrm{\mid }}_S$ of the quotient map whose images are vertical for every active map. Choose a locally finite smooth partition of unity ${\theta }_{\alpha }$ on $S$ subordinate to the neighbourhoods of step 1.1. Near each chart extend its vertical coordinate fields to the ambient chart. For a parameter $(s,v)\in Q$ let $W_{\alpha }(s,v;x)$ be the linear combination of these fields with the $Q$-coordinates of $v$ at $s$, multiplied by a smooth local cutoff in $(s,x)$ equal to one near the diagonal over the support of ${\theta }_{\alpha }$, and set $W(s,v;x)={\sum }_{\alpha }{\theta }_{\alpha }(s)W_{\alpha }(s,v;x)$. The sum is locally finite in $s$, smooth in all variables, and linear in $v$; at $x=s$ it equals $j_s(v)={\sum }_{\alpha }{\theta }_{\alpha }(s)j_{\alpha }(s)(v)$, whose quotient is $v$ because every summand splits the quotient map. Whenever $s\in C_i$, every summand of nonzero weight lies in a neighbourhood carrying that constraint, so $W(s,v;\cdot )$ is vertical for $F_i$ near $x=s$.

Let $E(s,v)$ be the time-one endpoint of the flow of $x\mapsto W(s,v;x)$ starting at $s$. By [F3] applied to the jointly smooth field $W$ this is defined and smooth on an open neighbourhood of the zero section: for small $v$ the field is small, and rescaling the parameter or restricting a common short-time interval gives the time-one solution. It satisfies $E(s,0)=s$ and $d_{v}E(s,0)=j_s$, because the field vanishes at $v=0$ and its first parameter derivative is the constant initial vector $j_s(v)$; moreover $d_{s}E(s,0)$ is the identity, so the differential of $E$ at the zero section is the isomorphism $(a,v)\mapsto a+j_s(v)$. Hence $E$ is a local diffeomorphism at every point of the zero section. If $s\in C_i$, a sufficiently small parameter keeps the whole trajectory in the region where $W$ is vertical for $F_i$, so $F_i(E(s,v))=F_i(s)$ there by the chain rule. All radius requirements so far are positive local requirements on $s$.

For $s\in S$ let $r(s)$ be the supremum of $a\in (0,1]$ such that $E$ is defined and injective on $D_a(s)=\{(t,v):t\in S,\mathrm{\parallel }t-s\mathrm{\parallel }<a,\mathrm{\parallel }v\mathrm{\parallel }<a\}$. It is positive: $E$ is a local diffeomorphism at each point of the zero section, and an injectivity neighbourhood of $(s,0)$ contains $D_a(s)$ for sufficiently small $a>0$ in a local bundle trivialization. If $a<r(s)$ then $D_a(s)$ is eligible by restriction, and $D_{a-\mathrm{\parallel }s-t\mathrm{\parallel }}(t)\subseteq D_a(s)$ shows $r(t)\ge r(s)-\mathrm{\parallel }s-t\mathrm{\parallel }$; exchanging $s$ and $t$ makes $r$ 1-Lipschitz. Choose a positive smooth radius $\delta (s)$ with $\delta (s)<r(s)\mathrm{/}16$, with $\mathrm{\parallel }E(s,v)-s\mathrm{\parallel }<r(s)\mathrm{/}16$ whenever $\mathrm{\parallel }v\mathrm{\parallel }<\delta (s)$, and with any prescribed positive size bounds enforced; such a smooth minorant exists by [F2], summing suitably small positive constants with a subordinate partition of unity on a locally finite cover on which the required local flow, identity and size bounds have positive lower bounds; the prescribed size bounds have this property by hypothesis.

Suppose $E(s,v)=E(t,w)$ with $d=\mathrm{\parallel }s-t\mathrm{\parallel }$ inside this radius domain. Then $d<(r(s)+r(t))\mathrm{/}16\le (2r(s)+d)\mathrm{/}16$, so $d<2r(s)\mathrm{/}15$; also $\mathrm{\parallel }v\mathrm{\parallel }<r(s)\mathrm{/}16$ and $\mathrm{\parallel }w\mathrm{\parallel }<r(t)\mathrm{/}16<(17\mathrm{/}240)r(s)$. Both parameter points therefore lie in the eligible domain $D_{r(s)\mathrm{/}2}(s)$, on which $E$ is injective, so $(s,v)=(t,w)$. After shrinking the radius further to stay inside the local-diffeomorphism locus, $E$ is an injective local diffeomorphism on its full variable-radius domain, hence a diffeomorphism onto an open neighbourhood $T$ of $S$; its inverse composed with the projection of $Q$ onto $S$ defines $\pi :T\to S$, the tube map is exactly $E$, and step 3.1 gives $F_i(x)=F_i(\pi (x))$ whenever $\pi (x)\in C_i$ and $x$ lies close enough to its zero-section point. The prescribed displacement and fiber-radius bounds were built into $\delta$ in step 4.1.