Let A and B be events with A$$ and B$$ being their complementary events. Given P$($AB$)=$ P$($A$$B$)$ and P$($AB$)=$ P$($A$$B$),$ prove that A and B are independent.

Note: You are to start with only what is given above then derive analytically, step by step, the end result that A and B are independent. You cannot assume an end result, which you have not proved yet, as truth and use such $$truth$$ in an intermediate step as part of your $$proof$.$ You cannot extrapolate from a specific instance used to verify that A and B are independent and characterize such a narrow verification of limited$-$scope as $$proof$$ of an universal truth, either.