e, let $B_0,  B_0,  \dots, B_\ell$ be the corresponding to $\Omega$ operational sequence of [bracket system]{}s, where $B_0$ is empty and $B_\ell$ is final. Let $B_{i^*}$ be the first [bracket system]{} of the sequence such that $B_{i^*} \cup \{ c \}$ is not a [bracket system]{}. The existence of $B_{i^*}$ follows from the facts that $B_0 \cup \{ c \}$ is a [bracket system]{} and $B_\ell \cup \{ c \}$ is not. Since $B_0 \cup \{ c \}$ is a [bracket system]{}, it follows that $