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

Below contains an error.

Trying to calculate (\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 (\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 \mathcal{A}, \mathcal{B}, \mathcal{C}.

Currently the proof is located in this file.

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