New theorem about funcoids (ERROR!)

I have proved (and added to my online book) the following theorem:

Theorem Let f \in \mathsf{FCD} (A ; B) and z \in \mathscr{F} (B)^A. Then there is an (obviously unique) funcoid g \in \mathsf{FCD} (A ; B) such that \langle g\rangle x = \langle f\rangle x for nontrivial ultrafilters x and \langle g\rangle @\{ p \} = z (p) for p \in A

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

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