Cauchy filters, a generalization for reloids

I have proved the theorem from Wikipedia that every Cauchy filter is contained in a maximal Cauchy filter (in fact I’ve proved a more general statement).

I don’t know the standard proof (and don’t know where to find it), so I’ve devised a proof myself. I hope (thanks to use the notation reloids in my proof) my proof may be much shorter (but less elementary) than the standard proof.

My proof is presented in this preliminary rough draft.