The modal logic S4.2 with the characteristic axioms 


4: $\square \alpha \rightarrow \square \square \alpha$


and


.2: $\lozenge \square \alpha \rightarrow \square \lozenge \alpha$


and


T: $\square \alpha \rightarrow \alpha$


is sound and complete for transitive, reflexive and connected frames. Such frames validate the closure principle


CP $\lozenge \square \alpha \wedge \lozenge \square \beta \rightarrow \diamond \square (\alpha \wedge \beta)$


Can someone help me with deriving CP in S4.2?



A question on the modal logic S4.2
$\let\B\Box\let\D\Diamond$
\begin{align\*}
\D\B\alpha\land\D\B\beta&\to\D\B\B\alpha\land\D\B\B\beta\\
&\to\B\D\B\alpha\land\D\B\B\beta\\
&\to\D(\D\B\alpha\land\B\B\beta)\\
&\to\D\D(\B\alpha\land\B\beta)\\
&\to\D\D\B(\alpha\land\beta)\\
&\to\D\B(\alpha\land\beta)
\end{align\*}
using the K-provable principle $\B p\land\D q\to\D(p\land q)$ and monotonicity of $\B$ and $\D$. Note that the axiom T is not needed.


