I have proved that for every funcoid $latex f:\prod A\rightarrow\prod B$ (where $latex A$ and $latex B$ are indexed families of sets) there exists a funcoid $latex \mathrm{Pr}^{(A)}_k f$ (subatomic projection) defined by the formula:

$latex \mathcal{X} \mathrel{\left[ \Pr^{\left( A \right)}_k f \right]} \mathcal{Y}
\Leftrightarrow \\
\prod^{\mathsf{RLD}}_{i \in \mathrm{dom}\, A}
\left( \left\{ \begin{array}{ll}
1^{\mathfrak{F} \left( A_i \right)} & \mathrm{if}\,
i \neq k ;\\
\mathcal{X} & \mathrm{if}\, i = k
\end{array} \right. \right) \mathrel{\left[ f \right]}
\prod^{\mathsf{RLD}}_{i \in \mathrm{dom}\, B} \left( \left\{
1^{\mathfrak{F} \left( B_i \right)} & \mathrm{if}\,
i \neq k ;\\
\mathcal{Y} & \mathrm{if}\, i = k
\end{array} \right. \right) . $

My draft book is modified to include this new theorem.

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