Dear Prof.Barber,

     This is the proof of the theorem I mentioned before. This proof is recorded in a book

  (titled as variational principle in elasticity and its applications)  written by Prof.Hai Chang Hu, one of the inventors of the well-known Hu-Washizu generalized variational principle. It is written in Chinese. I translated it into English. Of course, all errors and misunderstandings are due to me. I think this proof is elegant and accessible to most of the students with engineering background.

  Hope it helps

 best regards

 Xu Guo

The essential assumption of Cauchy Stress
is the existence of the limit  (\Delta f)/ (\Delta A) as  \Delta A-->0. 
This means that a infinitesimal small area can only sustain infinitesimal
small surface force. Therefore in the framework of conventional continuum
mechanics, we can only talk about the "force density (stress)" at a point.
The expression int_A (f.n)dA has no physical meaning. Only int_A (stress.n)dA
can give the total force exerting on a closed surface A.