how is nullity($AB$) - Nullity($B$) $\leq$ Nullity($A$)
leads to Rank($A$) + Rank($B$) $\leq$ Rank($AB$) + $n$
here's what i thought,
so rank-nullity theorm tells, rank($A$)+nullity($A$) = $n$, rank($B$)+nullity($B$)=$k$, and rank($AB$) +nullity($AB$) = $k$. but then if you move all the rank to the left, and nullity to the right side of an equation, i don't see how those two equalities are same.