\in B\}, \eta|_{Y} \right\rangle$ but $\S(Y) \nsubseteq \left\langle \{\beta |_{Y}:\beta \in B\setminus \{\gamma\}\}, \eta|_{Y} \right\rangle$ for any $\gamma \in B$ with $\ker \gamma \in \{\ker \beta : \beta \in \widetilde{B}\}$.

Suppose that the conditions $(i) - (iii)$ are satisfied. Then by Lemma \[le7\] we have $\left\langle \OP(X,Y),B\right\rangle =\T(X,Y)$. It remains to show that $B$ is minimal. Assume that there is $\gamma
\in B$ such that $\left\langle \OP(X,Y),B\setminus \{\gamma \}\right\rangle
=\T(X,Y)$. Note that $\alpha \beta |_{Y}=\alpha |_{Y}\beta |_{Y}$ for all $\alpha ,\beta \in \T(X,Y)$. Hence, we can conclude that $\S(Y)\subseteq
\left\langle \left\{ \beta |_{Y}:\beta \in \T(X,Y)\right\} \right\rangle
\subseteq \left\langle \left\{ \beta |_{Y}:\beta \in \OP(X,Y)\cup (B\setminus
\{\gamma \})\right\} \right\rangle =\left\langle \left\{ \beta |_{Y}:\beta
\in B\setminus \{\gamma \}\right\} ,\eta |_{Y}\right\rangle $ by Lemma \[lenew\]. Hence, $\ker \gamma \notin \{\ker\beta :\beta \in \widetilde{B}\}$ by $(iii)$. This implies that $\gamma \in B\setminus \widetilde{B}$ and $|(B\setminus \widetilde{B})\setminus \{\gamma \}|<|\mathcal{R}_{m}|$ by $(ii)$, i.e. $\mathcal{R}_{m}\nsubseteqq \{\ker \beta :\beta \in (B\setminus \widetilde{B})\setminus \{\gamma \}\}$. Since $\ker\gamma \notin \{\ker \beta :\beta \in \widetilde{B}\}$, we have $\mathcal{R}_{m}\nsubseteqq \{\ker \beta :\beta \in (B\setminus \{\gamma \})\}$ and by Lemma \[le9\], we obtain that $\left\langle \OP(X,Y),B\setminus \{\gamma
\}\right\rangle \neq \T(X,Y)$, a contradiction. This shows that $B$ is a minimal relative generating set of $\T(X,Y)$ 