Why do we subtract AI(T) when computing Fo(T) of bond futures


If anyone would please explain:

Why do we subtract Accrued Bond Interest at Time T when computing the futures price of a bond contract

Fo(T) = Quoted bond price + Accrued interest at time 0 - PV of all coupon income during contract life - Accrued Int. at time T

Why do we subtract accrued interest at time T?


Cause The contract expires between coupon payments