Formula for sum of horizontal and vertical differentials I am working through the book The Geometry of Jet Bundles by D.J. Saunders and came across the formula 
$$d_h+d_v = \pi_{k+1,k} ^\star \circ d,$$
which I am unable to prove. Here $d$ denotes the exterior derivative, $d_h$ and $d_v$ the horizontal and vertical differentials respectively, and $\pi_{k+1,k}^\star$ the pullback along $\pi_{k+1,k}$.
This statement might be completely trivial but I've been staring at it for the better part of two days now and I just can't get my head around it.

I am unsure of how standard the notation used in the book is, so I'll briefly introduce the relevant constructions.
We are considering a fiber bundle
$ \pi \colon E \longrightarrow M$
and its associated jet manifolds $J^k \pi$.
The relevant bundle projections are denoted
\begin{align*}\label{}
 \pi _k\colon J^k \pi & \longrightarrow M\\
 j^k_p\phi & \longmapsto p,\\\\
 \pi _{k,0}\colon J^k \pi & \longrightarrow E\\
 j^k_p\phi & \longmapsto \phi(p),\\\\
 \pi _{k,l}\colon J^k \pi & \longrightarrow J^l \pi \\
 j^k_p\phi & \longmapsto j^l_p \phi.
\end{align*}
and we denote the tangent and cotangent bundles by
\begin{align*}\label{}
 \tau_{J^k\pi}\colon TJ^k \pi & \longrightarrow J^k\pi,\\
 \tau^*_{J^k\pi}\colon T^*J^k \pi & \longrightarrow J^k\pi.
\end{align*}
The relevant construction starts with the fact that the pullback bundle
$ (\pi ^\star_{k+1,k}(TJ^k \pi) , \pi ^\star _{k+1,k}(\tau _{J^k \pi }),J^k \pi ) $
may be written as the direct sum of vertical and holonomic elements:
$$
 (\pi ^\star_{k+1,k}(V \pi _k )\oplus H \pi_{k+1,k}, 
 \pi ^\star _{k+1,k}(\tau _{J^k \pi }),
 J^k \pi ),
$$
and the corresponding decomposition of 
$ (\pi ^\star _{k+1,k}(T^*J^k \pi ),\pi ^\star _{k+1,k}(\tau ^*_{J^k \pi }),J^{k+1} \pi )$
into horizontal and contact elements:
$$
 (\pi ^\star_{k+1,k}(\pi ^\star _k(T^*M))\oplus C^* \pi _{k+1,k},
 \pi ^\star _{k+1,k}(\tau ^*_{J^k \pi }),
 J^{k+1} \pi ).
$$
Using these decompositions we can define the vector bundle endomorphisms $h$ and $v$:
Def:
    The vector bundle endomorphisms $ (h, \operatorname{id} _{J^k \pi }) $
    and $ (v, \operatorname{id} _{J^k \pi }) $ of $ \pi ^\star _{k+1,k}(\tau _{J^k \pi }) $
    are defined as
    \begin{align*}\label{}
  h(\xi ^h + \xi ^v) &= \xi ^h\\
  v(\xi ^h + \xi ^v) &= \xi ^v,
 \end{align*}
    where $ \xi^h \in H \pi _{k+1,k} $ and $ \xi ^v \in \pi ^\star _{k+1,k}(V \pi _k) $.
Def:
    The vector bundle endomorphisms $ (h, \operatorname{id} _{J^k \pi }) $
    and $ (v, \operatorname{id} _{J^k \pi }) $ of $ \pi ^\star _{k+1,k}(\tau^* _{J^k \pi }) $
    are defined as
    \begin{align*}\label{}
  h(\eta ^h + \eta ^v) &= \eta ^h\\
  v(\eta ^h + \eta ^v) &= \eta ^v,
 \end{align*}
    where 
    $ \eta ^h \in \pi ^\star _{k+1,k}(\pi ^\star_k(T^*M)) $
    and 
    $ \eta^v \in C^* \pi _{k+1,k} $.
The endomorphisms $h$ and $v$ now allow us to construct the following vector
valued 1-forms also called $h$ and $v$:
Def:
    The vector valued 1-forms $ h $ and $ v $ are the sections of the bundle
    $ \pi ^\star _{k+1,k}(\tau ^*_{J^k \pi })\otimes \pi ^\star _{k+1,k}(\tau _{J^k \pi }) $ 
    defined by
    \begin{align*}\label{}
  h_{j^{k+1}_p \phi }( \xi , \eta ) &= \eta (h(\xi ))\\
  v_{j^{k+1}_p \phi }( \xi , \eta ) &= \eta (v(\xi )),
 \end{align*}
    for $ \xi \in \pi ^\star _{k+1,k}(TJ^k \pi )_{j^{k+1}_p \phi } $ and 
    $ \eta \in \pi ^\star _{k+1,k}(T^*J^k \pi )_{j^{k+1}_p \phi } $.
The map
\begin{align}
 \pi ^\star_{k+1,k} (T^*J^k\pi) &\longrightarrow T^*J^{k+1}\pi\\
  (\eta , j^{k+1}_p \phi ) &\longmapsto (\pi_{k+1,k} ^\star \,\eta)_{j^{k+1}_p \phi} 
\end{align}
allows us to consider
$ \pi ^\star _{k+1,k}(\tau ^*_{J^k \pi })\otimes \pi ^\star _{k+1,k}(\tau _{J^k \pi }) $ 
as a subbundle of
$ \tau ^*_{J^{k+1} \pi }\otimes \pi ^\star _{k+1,k}(\tau _{J^k \pi }) $,
and so we shall regard $h$ and $v$ as vector-valued 1-forms along $\pi_{k+1,k}$.
Def:
If $\xi$ is a vector-valued 1-form along $\pi_{k+1,k}$, we define the interior multiplication by $\xi$ along $\pi_{k+1,k}$, denoted $\imath_\xi$, by
$$
 \imath _ \xi f =0 \text{ for all } f \in \mathcal{C} ^\infty(J^k\pi), 
$$
and 
$$
    (\imath_ \xi \vartheta) (X_1, \ldots, X_{s}):=
    \sum _\sigma (-1)^\sigma
    \vartheta \Big( \xi (X_{\sigma (1)}),
    \pi _{k+1,k\star}\, X_{\sigma (2)}, \ldots, \pi _{k+1,k\star} \,X_{\sigma (s)}\Big),
$$
for $ \vartheta \in \bigwedge ^s J^k\pi $  ($s \geq 1 $) and
            $ X_1, \ldots, X_{s} \in \mathfrak{X} (J^{k+1}\pi) $,
where we only sum over those permutations $\sigma$ that satsify
$\sigma(2)<\ldots<\sigma(s)$.
Having defined the interior multiplications $\imath_h$ and $\imath_v$ we
can now define the horizontal and vertical differentials:
\begin{align}
d_h &:= \imath_h \circ d - d \circ \imath_h,\\
d_v &:= \imath_v \circ d - d \circ \imath_v.\\
\end{align}

The relevant statement in the book

One consequence of the relationship $h+v= \pi _{k+1,k}$ is that $d_h +d_v = \pi _{k+1,k\star} \circ d$; this yields ...

can be found on page 216. I believe that the pushforward in this statement should in fact be the pullback but this is probably just a typo.
 A: I think I figured it out while typing up the question. The solution is indeed almost trivial and, as the author states, the only specific property of $h$ and $v$ necessary to prove the statement is
$$ h+v=\pi_{k+1,k\star}.$$
Since $d_h+d_v$ is a derivation along $\pi_{k+1,k}$ (this is easy to show)
and derivations are uniquely specified by how they act on functions and 1-forms
it suffices to check how $d_h+d_v$ acts on functions $f \in \mathcal{C}^\infty(J^k\pi)$ and 1-forms $\vartheta \in \bigwedge^1 J^k\pi$.
First notice that for 1-forms
$$
(\imath_h+\imath_v) \vartheta = \pi_{k+1,k}^\star \vartheta,$$
which gives
\begin{align}
(d_h+d_v)f&= (\imath_h+\imath_v)df\\
&= \pi_{k+1,k}^\star df,
\end{align}
where we used that $df$ is a 1-form and $\imath_hf=\imath_vf=0$.
    Now let $ X_0,X_1 \in \mathfrak{X}(J^{k+1}\pi) $, the definition of $ \imath_h $ simply gives
    $$
  (\imath_h d \vartheta )(X_0,X_1)
  = d \vartheta (hX_0, \pi _{k+1,k\star} X_1)
  - d \vartheta (hX_1, \pi _{k+1,k\star} X_0).
 $$
    Together with the analogous result for $ \imath_v $ this shows
    \begin{align*}\label{}
  ((\imath_h+\imath_v)d \vartheta )(X_0,X_1)
  &= d \vartheta (\pi_{k+1,k\star} X_0, \pi _{k+1,k\star} X_1)
  - d \vartheta (\pi_{k+1,k\star} X_1, \pi _{k+1,k\star} X_0)\\
  &= 2 (\pi _{k+1,k}^\star \, d \vartheta) (X_0,X_1),
 \end{align*}
    which already gives
    $$
  (d_h+d_v) \vartheta = \pi _{k+1,k}^\star\, d \vartheta.
 $$
So we have shown that $d_h+d_v$ and $\pi_{k+1,k}^\star\circ d$ agree on functions as well as on 1-forms, which, since both are derivations, implies that they are equal.
Is this correct?
