K4.3 is the normal logic axiomatized by 4 ($\Diamond \Diamond p \to \Diamond p$) and
$$
\Diamond p\land\Diamond q \to\Diamond(p\land\Diamond q)\lor\Diamond(p\land q)\lor\Diamond(q\land\Diamond p)\
$$
or by 4 and
$$
\Box(p\land\Box p \to q) \lor \Box(q\land\Box q \to p)\
$$

(These two forms of .3-axioms are given seperately and it remains to show they are equal. So it might be better to use only one of them in a proof.)

I have shown that the canonical modal for the modal logic S4.3 (given by $S4 + \Box(\Box p \to q) \lor \Box(\Box q \to p)$) has no branching to the right (the proof is similar to the one in [Show that the canonical modal for the modal logic s4.3 has no branching to the right](https://math.stackexchange.com/questions/1542553/show-that-the-canonical-modal-for-the-modal-logic-s4-3-has-no-branching-to-the-r)).

However I find it difficult to deal with the precondition $u \neq v$ , where w, u and v are points in the canonical model for K4.3 such that
$$
Rwu, Rwv, \lnot Ruv, \lnot Rvu, u \neq v.
$$
Since $\lnot Ruv$, there is $\varphi$ such that $ M,u \models \Box \varphi$ but $M,v \not \models \varphi$. Since $\lnot Rvu$, there is $\psi$ such that $ M,v \models \Box \psi$ but $M,u \not \models \psi$. And since $u \neq v$, there is $\chi$ such that $\chi\in u$ but $\lnot\chi \in v$.

It seems that I should consider three formulas $\varphi$, $\psi$ and $\chi$ rather than two. I've tried to use them to construct other two fprmulas, but found it difficult.

This question is in Blackburn, Patrick, Maarten de Rijke, and Yde Venema, 2001, *Modal Logic*, Cambridge: Cambridge University Press, p.211, Exercise 4.3.3.

Thank you so much! ^\_^