I have proved that for every funcoid 
![\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) .](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BX%7D+%5Cmathrel%7B%5Cleft%5B+%5CPr%5E%7B%5Cleft%28+A+%5Cright%29%7D_k+f+%5Cright%5D%7D+%5Cmathcal%7BY%7D++++%5CLeftrightarrow+%5C%5C+%5Cprod%5E%7B%5Cmathsf%7BRLD%7D%7D_%7Bi+%5Cin+%5Cmathrm%7Bdom%7D%5C%2C+A%7D++++%5Cleft%28+%5Cleft%5C%7B+%5Cbegin%7Barray%7D%7Bll%7D++++++1%5E%7B%5Cmathfrak%7BF%7D+%5Cleft%28+A_i+%5Cright%29%7D+%26+%5Cmathrm%7Bif%7D%5C%2C++++++i+%5Cneq+k+%3B%5C%5C++++++%5Cmathcal%7BX%7D+%26+%5Cmathrm%7Bif%7D%5C%2C+i+%3D+k++++%5Cend%7Barray%7D+%5Cright.+%5Cright%29+%5Cmathrel%7B%5Cleft%5B+f+%5Cright%5D%7D++++%5Cprod%5E%7B%5Cmathsf%7BRLD%7D%7D_%7Bi+%5Cin+%5Cmathrm%7Bdom%7D%5C%2C+B%7D+%5Cleft%28+%5Cleft%5C%7B++++%5Cbegin%7Barray%7D%7Bll%7D++++++1%5E%7B%5Cmathfrak%7BF%7D+%5Cleft%28+B_i+%5Cright%29%7D+%26+%5Cmathrm%7Bif%7D%5C%2C++++++i+%5Cneq+k+%3B%5C%5C++++++%5Cmathcal%7BY%7D+%26+%5Cmathrm%7Bif%7D%5C%2C+i+%3D+k++++%5Cend%7Barray%7D+%5Cright.+%5Cright%29+.+&bg=ffffff&fg=000&s=0&c=20201002 "\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.
Math Research of Victor Porton
Algebraic General Topology, Axiomatic Theory of Formulas, (generalized) limit of arbitrary function
I have proved that for every funcoid
My draft book is modified to include this new theorem.