Positive transverse accessibility is a preorder

Let ${\gamma }_A$ be a $C^2$ leafwise path from $x$ to $c(0)$ inside $A$ and ${\gamma }_B$ a $C^2$ leafwise path from $c(1)$ to $y$ inside $B$. Such paths exist: cover each leaf by its foliated charts, whose plaques are convex and chartwise polygonally connected, use [F2] finitely many times to reach the target plaque, and smooth the finitely many corners inside plaque charts; the finite plaque transport preserves the $C^2$ regularity of the leafwise curves by [F3]. Only finitely many charts and plaques are selected.

Near the compact union of the three curves ${\gamma }_A,c,{\gamma }_B$ construct a smooth field $X$ and a $C^1$ form $\omega$ with $\ker \omega =TF$ and $X$ positive transverse: choose finitely many smooth ambient chart bumps over that union by [F4], take in each chart a constant vector that is positive for the continuous cooriented tangent planes on a smaller neighbourhood, and sum the bumped fields; patch the local positively cooriented annihilating forms of the $C^2$ atlas in the same finite way. On a compact flow domain the smoothness of $X$ and $\omega$ gives a uniform bound $\omega (X)\ge b>0$ and, by [F5], a uniform interval $\mathrm{\mid }a\mathrm{\mid }\le a_0$ of the flow ${\phi }_a$ on which the derivative $D{\phi }_a$ is bounded; [F6] reduces the neighbourhood data to finitely many compact charts.

For a $C^2$ leafwise path $\gamma$ near the union and a $C^2$ offset profile $a(s)$, the curve $s\mapsto {\phi }_{a(s)}(\gamma (s))$ has transverse derivative $v(s,a(s))+a^{\mathrm{\prime }}(s)\omega (X)$, where $v(s,0)=\omega ({\gamma }^{\mathrm{\prime }}(s))=0$ because ${\gamma }^{\mathrm{\prime }}$ is tangent to $F$; hence $\mathrm{\mid }v(s,a)\mathrm{\mid }\le A\mathrm{\mid }a\mathrm{\mid }$ on the compact parameter set for a constant $A$ by [F7] applied to $a\mapsto v(s,a)$. With the profiles $a(s)=\delta (e^{Bs}-1)\mathrm{/}B$ on the initial path and $a(s)=-\delta (e^{B(1-s)}-1)\mathrm{/}B$ on the terminal path one has $a^{\mathrm{\prime }}=B\mathrm{\mid }a\mathrm{\mid }+\delta$, so choosing $B>A\mathrm{/}b$ gives transverse derivative at least $b\delta +(bB-A)\mathrm{\mid }a\mathrm{\mid }>0$; both profiles vanish at the outer endpoints, so ${\phi }_a{\gamma }_A$ starts at $x$ and ${\phi }_a{\gamma }_B$ ends at $y$.

Along $c$ use a fixed $C^2$ profile scaled by $\delta$ that joins the positive offset at its start to the negative offset at its end; since $\min \omega (c^{\mathrm{\prime }})>0$, its perturbed transverse derivative stays positive for small $\delta$, and the three pieces join into a piecewise $C^2$ positive path from $x$ to $y$. At each of the two internal seams choose one $C^2$ foliated chart; the one-sided transverse-coordinate derivatives have a positive lower bound there, and mollifying the continuous piecewise $C^2$ chart curve with a nonnegative kernel keeps the transverse derivative positive, while the blending correction has derivative tending uniformly to zero and is made smaller than the lower bound; the resulting curve is $C^2$, positive, agrees near the outer endpoint collars and still joins $x$ to $y$.

Transitivity: if genuine segments realize $A{⪰}_{F}B$ and $B{⪰}_{F}C$, apply the endpoint adjustment of step 3.1 to the second segment so that it starts at the actual endpoint of the first, concatenate the two, and smooth the single internal seam by the same chartwise mollification; the equality cases are immediate from the definition, so $A{⪰}_{F}C$.

Reflexivity holds by the defining equality clause of [F1], and transitivity is step 4.1, so ${⪰}_F$ is a preorder in the sense of [F8]; the endpoint conclusion follows from step 3.1 applied to any genuine joining segment, and the formal equality clause alone is never used to produce an actual segment.

Mutual accessibility is reflexive and symmetric by definition and transitive by two applications of the transitivity in step 4.1, so it is an equivalence relation in the sense of [F8]; the construction used only finitely many charts, bumps, flow intervals and profiles, hence only the standing countable choice from [F9].