respectively. This defines maps $$\begin{aligned}
 \Phi_{\mathbf{e}}: {\mbox{{\rm Gr}}_{{\mathbf{e}}}}(V\oplus W) & \longrightarrow & \coprod_{\mathbf{f} + {\mathbf{g}}= {\mathbf{e}}} {\mbox{{\rm Gr}}_{\mathbf{f}}}(V) \times {\mbox{{\rm Gr}}_{{\mathbf{g}}}}(W) \\
                 B & \longmapsto & (\iota^{-1}(B), \pi(B))\end{aligned}$$ which are constructible maps. These maps are clearly surjective. Moreover, the fiber of a point $(U_1, U_2)$ can be shown to be an affine space (it is isomorphic to ${\operatorname{{\rm Hom}}_{A}}(U_2, V/U_1)$, see [@CC06 Lemma 3.8]).

Thus, by Remark \[rema::Euler\], we get $$\begin{aligned}
 \chi\big( {\mbox{{\rm Gr}}_{{\mathbf{e}}}}(V\oplus W) \big) & = & \chi\big( \coprod_{\mathbf{f} + {\mathbf{g}}= {\mathbf{e}}} {\mbox{{\rm Gr}}_{\mathbf{f}}}(V) \times {\mbox{{\rm Gr}}_{{\mathbf{g}}}}(W) \big) \\
  & = & \sum_{\mathbf{f} + {\mathbf{g}}= {\mathbf{e}}} \chi\big( {\mbox{{\rm Gr}}_{\mathbf{f}}}(V) \big) \cdot \chi\big( {\mbox{{\rm Gr}}_{{\mathbf{g}}}}(W) \big).\end{aligned}$$

From there, the proof is a simple computation: $$\begin{aligned}
  F_V({\mathbf{y}})F_W({\mathbf{y}}) & = & \Big( \sum_{\mathbf{f}\in{\mathbb{N}}^{Q_0}} \chi\big( {\mbox{{\rm Gr}}_{\mathbf{f}}}(V) \big) {\mathbf{y}}^{\mathbf{f}} \Big) \cdot \Big( \sum_{{\mathbf{g}}\in{\mathbb{N}}^{Q_0}} \chi\big( {\mbox{{\rm Gr}}_{{\mathbf{g}}}}(W) \big) {\mathbf{y}}^{\mathbf{g}}\Big) \\
  & = & \sum_{\mathbf{f}, {\mathbf{g}}} \chi\big( {\mbox{{\rm Gr}}_{\mathbf{f}}}(V) \big) \chi\big( {\mbox{{\rm Gr}}_{{\mathbf{g}}}}(W) \big) {\mathbf{y}}^{\mathbf{f} + {\mathbf{g}}} \\
  & = & \sum_{\mathbf{e}}\Big( \sum_{\mathbf{f} + {\mathbf{g}}= {\mathbf{e}}} \chi\big( {\mbox{{\rm Gr}}_{\mathbf{f}}}(V) \big) \chi\big( {\mbox{{\rm Gr}}_{{\mathbf{g}}}}(W) \big)  \Big){\mathbf{y}}^{{\mathbf{e}}} \\
  & = & \sum_{\mathbf{e}}\chi\big( {\mbox{{\rm Gr}}_{{\mathbf{e}}}}(V\oplus W) \big) {\mathbf{y}}^{{\mathbf{e}}} \\
  & = & F_{V\oplus W}({\mathbf{y}}).\end{aligned}$$$\Box$

The second property of $F$-polynomials, and perhaps the most important one for our 