### Some new theorems

I added the following theorems to Funcoids and Reloids article. The theorems are simple to prove but are surprising, as do something similar to inverting a binary relation which is generally neither monovalued nor injective.

Proposition Let $latex f$, $latex g$, $latex h$ are binary relations. Then $latex g \circ f \not\asymp h \Leftrightarrow g \not\asymp h \circ f^{- 1}$.

Theorem Let $latex A$, $latex B$, $latex C$ are sets, $latex f \in \mathsf{FCD} (A ; B)$, $latex g \in \mathsf{FCD} (B ; C)$, $latex h \in \mathsf{FCD}(A ; C)$. Then

$latex g \circ f \not\asymp h \Leftrightarrow g \not\asymp h \circ f^{- 1} .$

Theorem Let $latex A$, $latex B$, $latex C$ are sets, $latex f \in \mathsf{RLD} (A ; B)$, $latex g \in \mathsf{RLD} (B ; C)$, $latex h \in \mathsf{RLD}(A ; C)$. Then

$latex g \circ f \not\asymp h \Leftrightarrow g \not\asymp h \circ f^{- 1} .$

The above theorems are the key for describing product funcoids, a task I previously got stuck. Now I can continue my research.