A second note on homological systems

Let $(\mathrm{\Delta };\mathrm{\Omega },\le )$ be a b-homological system and $\tilde{\mathcal{F}}\left(\mathrm{\Delta }\right)$ the category of the extended $\mathrm{\Delta }-$filtered modules. Here there is a proof that $\tilde{\mathcal{F}}\left(\mathrm{\Delta }\right)$ is closed under direct summands.