I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the theorem seems rather well-established.
So far, I have found dmoskovich's blog and a few papers by googling that are using the theorem as well.
Any pointers to a source that explicitely states and proves the above theorem are appreciated. I need the citation for my thesis, so I don't think I'm allowed to take any hints on how to actually prove the theorem :)