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\{

\begin{array}{ll}

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.