The proximal operator is defined by: $$ prox_{\tau f} = \arg\min_u \frac 1 {2\tau} \|u-u^k\|^2 + f(u) $$
I know the solution when $$ f(u) = \|Au+b\|_1 $$
But I wonder how to derive the solution of proximal operator when $$ f(u) = \|Au+b\|_1 + \|Bu+c\|_1$$