# Erroneous lemma corrected

In “Funcoids and Reloids” online draft there was an erroneous lemma:

Lemma For every two sets $S$ and $T$ of binary relations and every set $A$
$\bigcap {\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} T \Rightarrow \bigcap {\nobreak}^{\mathfrak{F}} \{ \langle F \rangle A | F \in S \} = \bigcap{\nobreak}^{\mathfrak{F}} \{ \left\langle G \right\rangle A | G \in T \}$.

The above lemma is false. The below modified lemma is true:

Lemma For every two filter bases $S$ and $T$ of binary relations and every set $A$
$\bigcap {\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} T \Rightarrow \bigcap {\nobreak}^{\mathfrak{F}} \{ \langle F \rangle A | F \in S \} = \bigcap{\nobreak}^{\mathfrak{F}} \{ \left\langle G \right\rangle A | G \in T \}$.

After correcting the lemma I corrected also the proof of the theorem which relies on this lemma:

Theorem $(\mathsf{FCD}) (g \circ f) = ((\mathsf{FCD} g)) \circ ((\mathsf{FCD}) f)$ for every reloids $f$ and $g$.