A proof for a theorem related to rank and matrix product.

Question

For all matrix $\mathbf{M} \in \mathbb{R}^{m,n}$ and $\mathbf{N} \in \mathbb{R}^{n,p}$, the inequality $\operatorname{rank}\mathbf{M} + \operatorname{rank}\mathbf{N} - n \leq \operatorname{rank}(\mathbf{M}\mathbf{N}) \leq \min(\operatorname{rank}\mathbf{M},\operatorname{rank}\mathbf{N})$ holds. What would be a proof of this theorem?