I'm going through an example proof, and I'm unclear why it's important to write: Let $\Gamma \cup \{\lnot B\}=A_1,...A_n,\lnot B$, why not let $\Gamma \cup \{\lnot B\}=\lnot B$.
By definition of $\Gamma=\lnot B \vDash_{taut} B, v(\lnot B), v(B)=\textbf{t}$ however this is in conflict of the definition of $F_{\lnot}(v(B))=\textbf{t}$ as $v(B)$ must $=\textbf{f}$.
Therefore either $v(\lnot B), v(B)$ will be $\textbf{f}$, and the statement is unsatisfiable?