Proposition

Proposition
P(E1|F)=P(E1|E2F)P(E2|F)+P(E1|E2cF)P(E2c|F)

If we define Q(E)=P(E|F), then Q(E) may be regarded as a probability function on the events of S. Hence all of the propositions previously proved for probabilities apply to it. For instance, we have

$Q(E_1\cup E_2)=Q(E_1)+Q(E_2)-Q(E_1E_2\vert F)$
or, equivalently,
$P(E_1\cup E_2\vert F)=P(E_1\vert F)+P(E_2\vert F)-P(E_1E_2\vert F)$
Also, if we define the conditional probability Q(E1|E2) by Q(E1|E2)=Q(E1E2)/Q(E2), then we have
$Q(E_1)=Q(E_1\vert E_2)Q(E_2)+Q(E_1\vert E_2^c)Q(E_2^c)\qquad\qquad(5.1)$
Since
$\begin{array}{rcl}
Q(E_1\vert E_2)&=&\displaystyle\frac{Q(E_1E_2)}{Q(E_2)} \\ \...
...(F)}}{\displaystyle\frac{P(E_2F)}{P(F)}} \\ \\
&=&P(E_1\vert E_2F)
\end{array}$
we see that Equation (5.1) is equivalent to P(E1|F)=P(E1|E2F)P(E2|F)+P(E1|E2cF)P(E2c|F)