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