 Given a structure \(M=\langle W,R_1,R_2,V\rangle\), the accessibility relation \(R_1\) is used to evaluate the box \(\square_1\), whereas \(R_2\) is used to evaluate \(\square_2\). Since the language of \({{\cal L}}\) is freely generated by the union of the signatures of \({{\cal L}}_1\) and \({{\cal L}}_2\), it contains mixed formulas such as \(\varphi = \square_1(\square_2 p \to p)\). Now, the structure \(M\) satisfies \(\varphi\) above at a world \(w \in W\) if and only if, for every \(w_1 \in W\) such that \(w R_1 w_1\), \(M\) satisfies \((\square_2 p \to p)\) at \(w_1\). But this means that, either there exists \(w_2\) such that \(w_1 R_2 w_2\) and \(w_2 \not\in V(p)\), or \(w_1 \in V(p)\).