The below is wrong! The proof requires 
I knew that composition of two complete funcoids is complete. But now I’ve found that for 
The proof which I missed for years is rather trivial:
![\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](https://s0.wp.com/latex.php?latex=%5Cbigsqcup+S+%5Cmathrel%7B%5Bg+%5Ccirc+f%5D%7D+J+%5CLeftrightarrow+J+%5Cmathrel%7B%5Bf%5E%7B-+1%7D+%5Ccirc+g%5E%7B-+1%7D%5D%7D+%5Cbigsqcup+S+%5CLeftrightarrow+%5Clangle+g%5E%7B-+1%7D+%5Crangle+J+%5Cmathrel%7B%5Bf%5E%7B-+1%7D%5D%7D+%5Cbigsqcup+S+%5CLeftrightarrow+%5Cbigsqcup+S+%5Cmathrel%7B%5Bf%5D%7D+%5Clangle+g%5E%7B-+1%7D+%5Crangle+J+%5CLeftrightarrow+%5Cexists+%5Cmathcal%7BI%7D+%5Cin+S+%3A+%5Cmathcal%7BI%7D+%5Cmathrel%7B%5Bf%5D%7D+%5Clangle+g%5E%7B-+1%7D+%5Crangle+J+%5CLeftrightarrow+%5Cexists+%5Cmathcal%7BI%7D+%5Cin+S+%3A+%5Cmathcal%7BI%7D+%5Cmathrel%7B%5Bg+%5Ccirc+f%5D%7D+J&bg=ffffff&fg=000&s=0&c=20201002 "\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
I will amend my book when (sic!) it will be complete.
Math Research of Victor Porton
Algebraic General Topology, Axiomatic Theory of Formulas, (generalized) limit of arbitrary function
The below is wrong! The proof requires
I knew that composition of two complete funcoids is complete. But now I’ve found that for
The proof which I missed for years is rather trivial:
Thus
I will amend my book when (sic!) it will be complete.
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