Derive Lipschitz norm equality.

Question

I am reading the paper "Spectral Normalization for Generative Adversarial Networks". The Lipschitz norm is defined as $$\|f\|_{Lip}=\max \frac{\|f(x)-f(x')\|}{\|x-x'\|}$$ In section 2.1, they claim that $\|g\|_{Lip}=\sup_h\sigma(\nabla g(h))$ while $\sigma$ is the spectral norm of the matrix. How can we prove this?