Decomposition of composition and a partial proof of a conjecture

Composition of binary relations can be decomposed into two operations: $latex \otimes$ and $latex \mathrm{dom}$:
$latex g \otimes f = \left\{ ( ( x ; z) ; y) \, | \, x f y \wedge y g z \right\}$.

Composition of binary relations is decomposed as: $latex g \circ f = \mathrm{dom} (g\otimes f)$.

I introduce similar decomposition of reloids, and using this try to prove that composition with a principal reloid is distributive over join of reloids.

The proof is partial, there are some white spots in it. The idea is very elegant, but I have failed to make a complete proof. Please email me or comment on this blog if you find a complete proof.

See this note about the proof (PDF).