Why B(tB) - A(tB) = AB

Why B(t_{_B}) - A(t_{_B}) = r_{_AB}?

Here's why -- the whole length B(t_{_B}) - A(t_{_A}) the photon travels along, is equal to the length of the rod r_{_AB} in the stationary system plus the length A(t_{_B}) - A(t_{_A}) which the beginning of the rod A covers during the time of travel of the light until it reaches B:

B(t_{_B}) - A(t_{_A}) = r_{_AB} + (A(t_{_B}) - A(t_{_A}))

B(t_{_B}) - A(t_{_B}) = r_{_AB}

or even simpler

B(t_{_B}) - A(t_{_B}) = [A(t_{_A}) + r_{_AB} + v(t_{_B} - t_{_A})] - [A(t_{_A}) + v(t_{_B} - t_{_A})] = r_{_AB}