If \( \{\alpha\alpha_i\} = \{\alpha_i'\} \), performing the right permutations, the sequence \( \alpha ,\alpha_1, \alpha_2, \dots, \alpha_n \) is identical with the sequence \( \alpha_1', \alpha_2', \dots, \alpha_{n+1}' \). This results in $$ \begin{equation} \langle\alpha_1\alpha_2 \dots \alpha_n|a_\alpha|\alpha\alpha_1\alpha_2 \dots \alpha_{n}\rangle = 1 \tag{18} \end{equation} $$ and thus $$ \begin{equation} a_\alpha |\alpha\alpha_1\alpha_2 \dots \alpha_{n}\rangle = |\alpha_1\alpha_2 \dots \alpha_{n}\rangle \tag{19} \end{equation} $$