a set $B\subseteq
\T(X,Y)$ to be a relative generating set of $\T(X,Y)$ modulo $\OP(X,Y)$.

\[le8\] Let $B\subseteq \T(X,Y)\setminus \OP(X,Y)$ with $\left\langle \OP(X,Y),B\right\rangle =\T(X,Y)$. Then $\S(Y)\subseteq \left\langle \{\beta |_{Y}:\beta \in B\},\eta|_{Y}\right\rangle$.

Let $\sigma \in \S(Y)$. We extend $\sigma$ to a transformation $\gamma : X \rightarrow Y$, i.e. $\gamma|_{Y}=\sigma$. Hence, there are $\gamma_{1},\ldots,\gamma_{k}\in \OP(X,Y)\cup B$ (for a suitable natural number $k$) such that $\gamma=\gamma_{1}\cdots \gamma_{k}$. Since the image of any transformation in $\T(X,Y)$ belongs to $Y$, we have $\sigma =\gamma|_{Y}=\gamma_{1}|_{Y}\cdots \gamma_{k}|_{Y}$. Moreover, from $\sigma \in \S(Y)$, we conclude $\gamma_{i}|_{Y}\in \S(Y)$ for $1\leq i\leq k$. Let $\gamma_{i}\in \OP(X,Y)$ for some $i\in\{1,\ldots,k\}$. Then by Lemma \[lenew\] $$\gamma_{i}|_{Y}=\left(
\begin{array}{cccccc}
a_{1} & \cdots  & a_{t} & a_{t+1} & \cdots  & a_{m} \\
a_{m-t+1} & \cdots  & a_{m} & a_{1} & \cdots  & a_{m-t}\end{array}\right) \in \langle\eta |_{Y}\rangle$$ for a suitable natural number $t$. This shows $\sigma
\in \left\langle \{\beta |_{Y}:\beta \in B\},\eta |_{Y}\right\rangle$.

\[le9\] Let $B\subseteq \T(X,Y)\setminus \OP(X,Y)$ with $\left\langle \OP(X,Y),B\right\rangle =\T(X,Y)$. Then $\mathcal{R}_{m}\subseteq \{\ker \beta :\beta \in B\}$.

Assume that there is $P\in \mathcal{R}_{m}$ with $P \not\in \{\ker \beta :\beta \in B\}$. Let $\gamma \in \T(X,Y)$ with $\ker\gamma =P$. Then there are $\theta_{1}\in \OP(X,Y)\cup B$ and $\theta_{2}\in \T(X,Y)$ such that $\gamma = \theta _{1}\theta _{2}$. Since 