this defining chain.

\[prop:DC\] Let $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. Then, $\pi \otimes \eta \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ if and only if there exists a defining chain for $\pi \otimes \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$.

We will give a proof of Proposition \[prop:DC\] in Section \[subsec:prf-DC\].

Now, let $\eta=(y_{1},\,\dots,\,y_{p}\,;\,{\mathbf{b}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. For each $x \in W_{{\mathrm{af}}}$ such that $\kappa(\eta) = y_{p} {\succeq}{\Pi^{K}}(x)$, we define a specific lift ${\iota(\eta,\,x)} \in W_{{\mathrm{af}}}$ of $\iota(\eta) = y_{1} \in ({W^{K}})_{{\mathrm{af}}}$ as follows. Since $y_{p} {\succeq}{\Pi^{K}}(x)$ by the assumption, it follows from Proposition \[prop:Deo\] that ${\mathrm{Lift}_{{\succeq}x}(y_{p})}$ has the minimum element $\min {\mathrm{Lift}_{{\succeq}x}(y_{p})}=:{\widetilde{y}}_{p}$. Similarly, since $y_{p-1} {\succeq}y_{p} = {\Pi^{K}}({\widetilde{y}}_{p})$, it follows again from Proposition \[prop:Deo\] that ${\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{p}}(y_{p-1})}$ has the minimum element $\min {\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{p}}(y_{p-1})} = :{\widetilde{y}}_{p-1}$. Continuing in this way, we obtain ${\widetilde{y}}_{p}$, ${\widetilde{y}}_{p-1}$, $\dots$, ${\widetilde{y}}_{1}$. Namely, these elements are defined by the following recursive procedure (from $p$ to $1$): $$\label{eq:lift0}
\begin{cases}
{\widetilde{y}}_{p} := \min {\mathrm{Lift}_{{\succeq}x}(y_{p})}, & \\[1mm]
{\widetilde{y}}_{q} := \min {\mathrm{Lift}_{{\succeq}{\widetilde{y}}_{q+1}}(y_{q})}
& \text{for $1 \le q \le p-1$}.
\end{cases}
$$ Finally, we set $$\label{eq:lift0a}
{\iota(\eta,\,x)}:={\widetilde{y}}_{1};$$ this element is sometimes called the initial direction of $\eta$ with respect to $x$.

\[prop:DC2\] Let $\pi \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$. Then, $\pi \otimes \eta \in {\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ [(]{}or equivalently, 