ECKS-PS algorithm: searching in encrypted data; bilinear maps
Note I have changed the value of $H(W_2)$ to make intermediate caclulation results unique.
$e(g^i,g^j) =g^{i*j}\mod 59,g=53,\alpha=7,r=16,\theta=3,a_i=6,H(W_1)=53^{23},H(W_2)=53^{19}$

$e(g^{a_i \theta},g^r)=53^{6*3 * 16} \mod 59=53^{27}=41$

$A=(H(W_1)*H(W_2))^\alpha . g^{\theta r}=53^{(23+19)*7+3*16 \mod 29}=53^{23}$

$e(g^{a_i},A) = 53 ^ {6*23} \mod 59 =  28$

$e(g^{\alpha}, H(W_1)^{a_i}) = 53^{7 * 23*6} \mod 59 = 53^9 \mod 59 = 35$

$e(g^{\alpha}, H(W_2)^{a_i}) = 53^{7 * 19*6} \mod 59 = 53^{15} \mod 59 = 17$

$\frac {28}{35*17} \mod 59= 41$ QED