theorem :: PROB_2:37

for Omega being non empty set
for Sigma being SigmaField of Omega
for B, A1, A2 being Event of Sigma
for P being Probability of Sigma st 0 < P . B & A2 = A1 ` & 0 < P . A1 & 0 < P . A2 holds
( (P .|. B) . A1 = (((P .|. A1) . B) * (P . A1)) / ((((P .|. A1) . B) * (P . A1)) + (((P .|. A2) . B) * (P . A2))) & (P .|. B) . A2 = (((P .|. A2) . B) * (P . A2)) / ((((P .|. A1) . B) * (P . A1)) + (((P .|. A2) . B) * (P . A2))) )