\eta $ and $\rank\theta _{i}=m$ for $i\in \{1,\ldots ,l\}$ and thus, $\{1,n\}\subseteq Y$ implies $(1,n)\notin \ker \theta _{i}$ for $i \in
\{2,\ldots ,l\}$. This implies $\theta _{2},\ldots ,\theta _{l}\in \O(X,Y)$. Since $\ker \theta _{1}=\ker \eta \notin \mathcal{P}_{m}$, we get $\theta
_{1}\in \O(X,Y)$, and consequently, $\eta =\theta _{1}\theta _{2}\cdots
\theta _{l}\in \O(X,Y)$, a contradiction. So, we have verified that $\left\vert \mathcal{P}_{m}\right\vert <\left\vert B\right\vert $, i.e. the relative rank of $\OP(X,Y)$ modulo $\O(X,Y)$ is greater than $\left(
\begin{array}{c}
n-1 \\
m\end{array}\right) $. But it is bounded by $1+\left(
\begin{array}{c}
n-1 \\
m\end{array}\right)$ due to Proposition \[le2\]. This proves the assertion.

  \
We finish this section with the characterization of the minimal relative generating sets of $\OP(X,Y)$ modulo $\O(X,Y)$. We will recognize that among them there are sets with size greater than $\rank(\OP(X,Y):\O(X,Y))$.

Let $B\subseteq \OP(X,Y)$. Then $B$ is a minimal relative generating set of $\OP(X,Y)$ modulo $\O(X,Y)$ if and only if for the set $\widetilde{B}=\{\beta
\in B:\ker \beta \in \mathcal{Q}_{m}\} \subseteq B$ the following three statements are satisfied:$(i)$ $\mathcal{P}_{m} \subseteq \{\ker \beta :\beta \in B\setminus \widetilde{B}\}$,$(ii)$ $|B\setminus \widetilde{B}| = |\mathcal{P}_{m}|$, $(iii)$ $\eta |_{Y}\in \left\langle \beta |_{Y}:\beta \in B\right\rangle $ but $\eta |_{Y}\notin \left\langle \beta |_{Y}:\beta \in B\setminus \{\gamma
\}\right\rangle $ for any 