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Jacobian and level curve

This problem is from 'Introduction to Mathematical Analysis' by Douglass.

Problem 9.15 Let f=(f_1,f_2) be a continuously differentiable function defined on an open set U in R^2 such that grad(f_1) and grad(f_2) do not vanish at any point of U. Suppose that jacobian of f is zero for all x in U. Prosve that a curce C in U is a level curve of f_1 if and only if it is also a level curve of f_2.

I don't know how to use jacobian condition to prove statement.