**

Either $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$ or $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$.

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Then, if a pair of operators $a$ and $b$ acting on ${\cal H}$ satisfies Assumption 1-3, $( \{ \phi_{n} \} , \{ \psi_{n} \} )$ becomes a semi-regular biorthogonal pair.

By Section 2, Section 3 and Ref. [@hiroshi1], in case of $D(\phi)$ and $D_{\phi}$ are dense in ${\cal H}$ (resp. $D(\psi)$ and $D_{\psi}$ are dense in ${\cal H}$), $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$ and $N_{\phi,\bm{e}}$ are lowering, raising and number operators for $\{ \phi_{n} \}$, respectively, and $A_{\bm{e},\phi}$, $B_{\bm{e},\phi}$ and $N_{\bm{e},\phi}$ are raising, lowering and number operators for $\{ \psi_{n} \}$, respectively. (resp. $A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$ and $N_{\psi,\bm{e}}$ are lowering, raising and number operators for $\{ \psi_{n} \}$, respectively, and $A_{\bm{e},\psi}$, $B_{\bm{e},\psi}$ and $N_{\bm{e},\psi}$ are raising, lowering and number operators for $\{ \phi_{n} \}$, respectively.). And we have $$\begin{aligned}
A_{\phi,\bm{e}}B_{\phi,\bm{e}} - B_{\phi,\bm{e}}A_{\phi,\bm{e}} \subset I \;\;\; &{\rm  and}& \;\;\;
B_{\bm{e},\phi} A_{\bm{e},\phi} -A_{\bm{e},\phi}B_{\bm{e},\phi}\subset I . \nonumber \\
({\rm resp.} \;\;\; A_{\psi,\bm{e}}B_{\psi,\bm{e}} - B_{\psi,\bm{e}}A_{\psi,\bm{e}} \subset I \;\;\; &{\rm  and}& \;\;\;
B_{\bm{e},\psi} A_{\bm{e},\psi} -A_{\bm{e},\psi}B_{\bm{e},\psi}\subset I .) \nonumber \end{aligned}$$ Furthermore, we have the following statements with respect to the operators $A_{\phi,\bm{e}}$, $B_{\phi,\bm{e}}$, $A_{\bm{e},\phi}$ and $B_{\bm{e},\phi}$ (resp. $A_{\psi,\bm{e}}$, $B_{\psi,\bm{e}}$, 