Studying chromatic homotopy theory I encountered the chromatic fracture square $\require{AMScd}$ \begin{CD} L_{K(n) \vee K(m)}X @>>> L_{K(m)}X\\ @V V V @VV V\\ L_{K(n)}X @>>> L_{K(n)}L_{K(m)}X \end{CD} for $n<m$. Then I was told that applying a Bousfield localization functor we preserve the homotopy pull-back, this is useful since applying $L_{K(t)}$ for $n <t<m$ we get the equality $L_{K(t)}L_{K(n)\vee K(m)}X= L_{K(t)}L_{K(m)}X $ since the lower terms become zero.
I would like to know a proof of this claim. Also a reference for this fact is well accepted. Thank you for any help.
I just need to observe that the Bousfield localization, being a left adjoint, preserves homotopy colimits. Now by stability of the stable homotopy category homotopy pull-backs and pushouts coincide.