The question then becomes: does a real-valued harmonic function "u" exist that is defined on all of "U" and has the given boundary condition? The positive answer is provided by the Dirichlet principle. Once the existence of "u" has been established, the Cauchy–Riemann equations for the holomorphic function "g" allow us to find "v" (this argument depends on the assumption that "U" be simply connected). Once "u" and "v" have been constructed, one has to check that the resulting function "f" does indeed have all the required properties.