I have a problem with comparing two lower confidence intervals of odds ratio. The detail is follow:

Given: a,b,c >=1 ; d >=2; a+b = D1; c+d = D2 (D1 and D2 are constant)

lci1 = e^[ln(a*d/b*c)-1.96*square root(1/a+1/b+1/c+1/d)]

lci2 = e^[ln(a*(d-1)/b*(c+1) - 1.96*square root(1/a+1/b+1/(c+1)+1/(d-1)]

I want to approve lci1 > lci2 with all a,b,c,d

I have progammed this problem in computer. As the result, it is true ( lci1 > lci2 ). We want demonstrate this inequation in mathematical style. Could you please help me?

Thank you in advance!

Best,