Superposition de calques monochromes d'opacit\'es variables

For a monochrome layer $x$ of opacity $0\le o_x\le 1$ placed on another monochrome layer of opacity 1, the result given by the standard formula is $$\mathrm{\Pi }\left({\mathbf{C}}_{\phi }\right)=1+\sum _{n=1}^2(2-n-(-1{)}^n{o}_{\chi (\phi +1)})(\chi (n+\phi -1)-o_{\chi (n+\phi -1)}),$$ the formula being of course explained in detail in this paper. We will eventually deduce a very simple theorem, generalize it and then see its validity with alternative formulas to this standard containing the same main properties here exposed.