A new partial result about products of filters [ERROR!]

Below contains an error.

Trying to calculate $latex (\mathcal{B} \times^{\mathsf{RLD}}_F \mathcal{C}) \circ (\mathcal{A}
\times^{\mathsf{RLD}}_F \mathcal{B})$, I’ve proved (not yet quite thoroughly checked for errors) the following partial result:

Proposition $latex (\mathcal{B} \times^{\mathsf{RLD}}_F \mathcal{C}) \circ (\mathcal{A}
\times^{\mathsf{RLD}}_F \mathcal{B}) \neq \mathcal{A}
\times^{\mathsf{RLD}}_F \mathcal{C}$ for some proper filters $latex \mathcal{A}$, $latex \mathcal{B}$, $latex \mathcal{C}$.

Currently the proof is located in this file.