# Some new theorems

I’ve proved some new theorems. The proofs are currently available in this PDF file.

Theorem The set of funcoids is with separable core.

Theorem The set of funcoids is with co-separable core.

Theorem A funcoid $f$ is complete iff
$f = \bigsqcap^{\mathsf{FCD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, | \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in A : \langle T \rangle^{\ast} \{ x \} \in G (x) \right\}$.

Theorem A reloid $f$ is complete iff
$f = \bigsqcap^{\mathsf{RLD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f} (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, | \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in A : \langle T \rangle^{\ast} \{ x \} \in G (x) \right\}$.

It seems (I have not yet checked) that the following conjecture follows from the last theorem:

Conjecture Composition of complete reloids is complete.