if its restriction to every line parallel to has (one-dimensional) Lebesgue measure zero. Considering in particular the set in where the -directional derivative of fails to exist (which must be proved to be measurable), the latter condition is met due to the one-dimensional case of Rademacher's theorem.

The second step of Morrey's proof establishes the linear dependence of the -directional derivative of upon . This is based upon the following identity:
:Using the Lipschitz assumption on , the dominated convergence theorem