Prove that $\nabla \times (\nabla \times \vec A) = \nabla(\nabla \cdot \vec A) - \nabla^2\vec A$. For any vector function of $\vec A$.

Question

Show that $\nabla \times (\nabla \times \vec A) = \nabla(\nabla \cdot \vec A) - \nabla^2\vec A$. For any vector function of $\vec A$.