Suppose $u \in C^2(\R^n)$  is a harmonic function.  Prove that $v=|\nabla u|^2$ is subharmonic, i.e. $-\Delta v \leq 0$

Suppose $u \in C^2(\R^n)$ is a harmonic function. Prove that $v=|\nabla u|^2$ is subharmonic, i.e. $-\Delta v \leq 0$

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