New theorem about funcoids (ERROR!)

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

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

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