The proof was with an error (see the previous edited post)!

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I knew that composition of two complete funcoids is complete. But now I’ve found that for to be complete it’s enough to be complete.

The proof which I missed for years is rather trivial:

Thus is complete.

I will amend my book when (sic!) it will be complete.

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**Proposition** For , a finite set and a function there exists (obviously unique) such that for and for .

This funcoid is determined by the formula

and it corollary:

**Corollary** If , , and , then there exists an (obviously unique) such that for all ultrafilters except of and .

This funcoid is determined by the formula

.

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The error was that I claimed that infimum of a greater set is greater (while in reality it’s lesser).

I will delete the erroneous theorem from my book soon.

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Alleged counter-example:

and for infinite sets and .

I am now attempting to locate the error in the proof.

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**Theorem** Let and . Then there is an (obviously unique) funcoid such that for nontrivial ultrafilters and for

After I started to prove it, it took about a hour or like this to finish the proof.

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The proof is currently available in the section “Some inequalities” of this PDF file.

Note that earlier I put online some erroneous proof related to this.

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The proof is currently available in the section “Some inequalities” of this PDF file.

The proof isn’t yet thoroughly checked for errors.

Note that I have not yet proved , but the proof is expected to be similar to the above.

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