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

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

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

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

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

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