\begin{multline}\label{eqn:Y'_i}
  Y'_i = B'_i\cup\{\alpha_i\}=(Y_0\cap B_i)\cup\{\alpha_i\}=(Y_0\cup\{\alpha_i\})\cap (B_i\cup\{\alpha_i\})=Y_0\cap Y_i.%=(Y_0\cap B_i)\cup Y_0\cap\{\alpha_i\}\\
\end{multline}
