The below is wrong! The proof requires $latex \langle g^{-1}\rangle J$ to be a principal filter what does not necessarily hold.
I knew that composition of two complete funcoids is complete. But now I’ve found that for $latex g\circ f$ to be complete it’s enough $latex f$ to be complete.
The proof which I missed for years is rather trivial:
$latex \bigsqcup S \mathrel{[g \circ f]} J \Leftrightarrow J \mathrel{[f^{- 1} \circ
g^{- 1}]} \bigsqcup S \Leftrightarrow \langle g^{- 1} \rangle J \mathrel{[f^{-
1}]} \bigsqcup S \Leftrightarrow \bigsqcup S \mathrel{[f]} \langle g^{- 1}
\rangle J \Leftrightarrow \exists \mathcal{I} \in S : \mathcal{I}
\mathrel{[f]} \langle g^{- 1} \rangle J \Leftrightarrow \exists \mathcal{I}
\in S : \mathcal{I} \mathrel{[g \circ f]} J$
Thus $latex g\circ f$ is complete.
I will amend my book when (sic!) it will be complete.
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