I hope someone can help me with this little exercise (Exercise 4.1,Probability theory, Jaynes):

Suppose that we have vectors of events \{H_1,...,H_n\} and \{D_1,...,D_m\} which satisfy:

(1) P(H_i H_j)=0 for any i\neq j and \sum_iP(H_i)=1

(2) P(D_rD_s|H_i)=P(D_r|H_i)P(D_s|H_i), for any r\neq s, 1\leq i\leq n

(3) P(D_rD_s|\overline{H_i})=P(D_r|\overline{H_i})P(D_ s|\overline{H_i}), for any r\neq s, 1\leq i\leq n

where \overline{H_i} means the negation of H_i.


Prove: If n>2, then at most one of the following fractions

\frac{P(D_1|H_i)}{P(D_1|\overline{H_i})},\frac{P(D _2|H_i)}{P(D_2|\overline{H_i})},...,\frac{P(D_m|H_ i)}{P(D_m|\overline{H_i})} can differ from unity, 1\leq i\leq n.