I'm currently reading Introduction to h-Principle by Eliashberg. There the author makes a comment that the space of holonomic extensions is a convex space. To elaborate, consider $\pi:X\to V$ to be a smooth fibration. $A\subset V$ be an arbitrary subset. A section $F:A\to X^{(r)}$ is said to be holonomic if there exists a holonomic section $\tilde{F}:Op(A)\to X^{(r)}$, such that $\tilde{F}|_A=F$. Here $X^{(r)}$ is the $r^\text{th}$ jet space and $Op(A)$ is an open nbd of $A$ which can be chosen suitably.
Now the problem is to produce a continuous family holonomic sections defined on some $Op(A)$ that connects any two extensions of the same $F$. If the fibration was in fact a vector bundle, this is easy : linear homotopy along the fibers work. But how to justify this for arbitrary fibrations?
One idea was to use tubular neighborhoods about the image of the sections (any section is after all an injective immersion), but I'm getting stuck.
Any help is appreciated!