  For a monochrome layer $x$ of opacity $0\le o_x\le1 $ placed on another
monochrome layer of opacity 1, the result given by the standard formula is
$$\small\Pi\left({\bf
C}_\varphi\right)=1+\sum_{n=1}^2\left(2-n-(-1)^no_{\chi(\varphi+1)}\right)\left(\chi(n+\varphi-1)-o_{\chi(n+\varphi-1)}\right),$$
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.
