Let $A = FHG$ where $F$ has full column rank and $G$ has full row rank. Show that $rank(A) = rank(H)$

Question

The hint given was that for $A\in\mathbb{C}^{m\times n}_{r}, A^{(1)}A = I_n \iff r= n$ and $AA^{(1)} = I_m \iff r = m$ where $A^(1)$ is the generalized {1}- inverse (satisfies eq. 1 of the Moore-Penrose inverse)