# If a reloid is both complete and co-complete it is discrete

Today I proved the following conjecture:

If a reloid is both complete and co-complete it is discrete.

The proof was easily constructed by me shortly after I noticed an obvious but not noticed before proposition:

Proposition A reloid $f$ is complete iff there exists a function $G : \mho \rightarrow \mathfrak{F}$ such that
$f = \bigcup^{\mathsf{RLD}} \{ \{\alpha\} \times^{\mathsf{RLD}} G (\alpha) | \alpha \in \mho \}$.

Note that a similar theorem holds for funcoids.

See the online article for details about funcoids and reloids.