A draft proof of distributivity of composition with a principal reloid over join of reloids

Recently I’ve announced that I have an elegant proof idea of this conjecture, but have a trouble to fill in details of the proof:

Statement Composition with a principal reloid is distributive over join of reloids.

Now I have almost complete draft proof of the above statement (Well, it is yet to be checked for errors and some details to be filled in.)

Using this idea I hope to turn the last proved result into a stronger result (about distributivity of composition with a complete reloid, not necessarily principal, over join of reloids).