be decomposed into two convex sets, which contain $1$ and $n$, respectively. Since $k\leq m$, we can divide the partition $\ker \gamma$ into a partition $P\in \mathcal{R}_{m}$ such that $P$ contains a non-convex set which cannot be decomposed into two convex sets, which contain $1$ and $n$, respectively (if $k=m$ then we put $P=\ker \gamma$). Since $\mathcal{R}_{m}\subseteq \{\ker
\beta :\beta \in B\}$, there is $\lambda \in B$ with $\ker \lambda = P$. It is clear that $X\lambda = Y$.\
Further, let $X\gamma = \{y_1 < y_2 < \cdots < y_k\}$ and define the sets $$A_i = \{x \in Y : x\lambda^{-1} \subseteq y_i\gamma^{-1}\}$$ for $i = 1,\ldots,k$. It is clear that $\{A_1, A_2, \ldots, A_k\}$ is a partition of $Y$. Moreover, let $\{C_1 < C_2 < \cdots < C_k\} \in \mathcal{Q}_k$ be a partition of $X$ such that $|C_i \cap Y| = |A_i|$ for all $i = 1,\ldots,k$. Let $A_i = \{a_{i_1} < a_{i_2} < \cdots < a_{i_{t_i}}\}$ and $C_i \cap Y = \{c_{i_1} < c_{i_2} < \cdots < c_{i_{t_i}}\}$ with $t_i \in \{1, \ldots, m\}$ for $i \in \{1, \ldots, k\}$. We define a bijection $$\sigma: \bigcup_{i=1}^k A_i = Y \longrightarrow \bigcup_{i=1}^k (C_i \cap Y) = Y$$ on $Y$ 