The images of the maximal invariant sets of the two plugs are invariant under the time-t maps of the glued vector field

Let $(U,X)$ and $(V,Y)$ be plugs: compact Hausdorff smooth 3-manifolds with boundary carrying nonsingular C¹ vector fields transverse to the boundary. Let $T^{out}\subseteq\partial^{out}U$ and $T^{in}\subseteq\partial^{in}V$ be unions of connected components of the exit and entrance boundaries, and let $\varphi:T^{out}\to T^{in}$ be a C¹ diffeomorphism (the mission's IsBoundaryDiffeo). Let $(W,Z,i_U,i_V)$ be a plug gluing (the mission's IsPlugGluing): $W$ is a compact Hausdorff smooth 3-manifold with boundary, $i_U:U\to W$ and $i_V:V\to W$ are C¹ embeddings with injective derivatives whose images cover $W$ and meet exactly along the seam $i_U(x)=i_V(\varphi(x))$, $x\in T^{out}$, and $Z$ is the vector field on $W$ with $Z\circ i_U=Di_U\circ X$ and $Z\circ i_V=Di_V\circ Y$. The field $Z$ is not assumed to be C¹. We write $\phi^Z_t$ for the time-$t$ map of $Z$, the mission's flowMap Z t: it sends $y$ to $\gamma(t)$ for a chosen integral curve $\gamma$ of $Z$ on $[0,t]$ with $\gamma(0)=y$ when one exists (the mission's FlowDefined Z y t), and to $y$ otherwise. Let $\Lambda_X$ and $\Lambda_Y$ be the maximal invariant sets of $X$ and $Y$ (the points that lie on an integral curve defined for all times). Then the two pieces are invariant under the time-$t$ maps of $Z$:

In words: if $\delta$ is an integral curve of $X$ defined for all times, then $i_U\circ\delta$ is an integral curve of $Z$ defined for all times, and by uniqueness of integral curves of $Z$ (plugGluing_integralCurveOn_unique) the time-$t$ map of $Z$ sends $i_U(\delta(0))$ to $i_U(\delta(t))$. It is a step of the mission's proof of Proposition 1.1 (Section 3.1 of arXiv v1, p. 14) that the paper takes for granted. The invariance of the bundles of the two pieces under the flow of $Z$ refers to these maps.

Formalization Note The hypotheses are those of the mission's gluing statements. The proof in the mission uses the plug hypotheses, hTout, hTin and the hypothesis on $\varphi$ only through the uniqueness theorem plugGluing_integralCurveOn_unique. The statement itself needs uniqueness of integral curves of $Z$ only along orbits that stay in the images of the interiors of $U$ and $V$, away from the seam; the hypotheses are kept so that they match the mission's gluing statements. The plugs are not assumed hyperbolic.