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.

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