Inward-pointing fields have local forward semiflows at the boundary

Extend the coordinate components of $X$ across the face of a boundary chart at $p$. By [L2] the extension has a Euclidean local flow, and by [L1] its last component is positive at $p$.

By continuity, shrink to an ambient coordinate neighbourhood $W$ on which the last component of the extended field is at least some $c>0$, and shrink the initial neighbourhood and time so that all relevant trajectories remain in $W$. Let $h(t)$ be the last coordinate of a forward trajectory with $h(0)\ge 0$. If $h(t_1)<0$ for some $t_1>0$, let $\tau$ be the largest zero of $h$ in $[0,t_1]$; it exists by continuity. Then $h<0$ on $(\tau ,t_1]$, while $h^{\mathrm{\prime }}(t)\ge c$ there, so the one-variable mean-value theorem gives $h(t_1)-h(\tau )>0$, contradicting $h(t_1)<0=h(\tau )$. Thus the restricted forward flow remains in $M$; no analogous negative-time conclusion follows.