# Two similar theorems about funcoids and reloids

I proved the following two similar theorems about funcoids and reloids:

Theorem

1. For a complete funcoid $f$ there exist exactly one function $F \in \mathfrak{F}^{\mho}$ such that $f = \bigcup^{\mathsf{FCD}} \left\{ \{ \alpha \} \times^{\mathsf{FCD}} F(\alpha) | \alpha \in \mho \right\}$.
2. For a co-complete funcoid $f$ there exist exactly one function $F \in \mathfrak{F}^{\mho}$ such that $f = \bigcup^{\mathsf{FCD}} \left\{ F(\alpha) \times^{\mathsf{FCD}} \{ \alpha \} | \alpha \in \mho \right\}$.

Theorem

1. For a complete reloid $f$ there exist exactly one function $F \in \mathfrak{F}^{\mho}$ such that $f = \bigcup^{\mathsf{RLD}} \left\{ \{ \alpha \} \times^{\mathsf{RLD}} F(\alpha) | \alpha \in \mho \right\}$.
2. For a co-complete reloid $f$ there exist exactly one function $F \in \mathfrak{F}^{\mho}$ such that $f = \bigcup^{\mathsf{RLD}} \left\{ F(\alpha) \times^{\mathsf{RLD}} \{ \alpha \} | \alpha \in \mho \right\}$.

See this online article for definitions of used concepts and proofs.