Reproving a known theorem in an article Suppose one wants to use a theorem that was published quite a long time ago (+80 years) in a paper that is using a terminology and notations that are very much out-dated (making the paper very hard to read). Is it okay if we want to reformulate the result as well as the proof in an article using a more modern language (of course giving credit to the original paper)?
Or, on the other hand, is it something one shouldn't put in a research paper, since it's not a new result ? What is the general policy for this kind of things ?   
 A: I often go even further and reprove even "well-known" things if I cannot find a reference to my liking (= well-written and accessible for free and stated close enough to how I want to use it and ...). My common sense is that for a reader (and most of our readers are graduate students that normally do not have extensive knowledge of the subject but rather want to taste it for the first time), it is much easier to skip a few pages (normally I put such things in the Appendix, so even the word "to skip" is an exaggeration of the required effort) than to drive to the library or to have a terrifying moral dilemma of whether to pay 25 bucks to the copyright holders or to commit the unspeakable crime of piracy for the umpteenth time.
In other words, when (or, rather, if) I bother to write at all, my primary concern is the convenience of the reader and everything else is secondary and should be taken care of only if and only to the extent that it doesn't interfere with the main goal. The only "no-no" is knowingly making an (implicit) impression that you claim something already done as your own result. Almost every other restraint is an invention of crooks that know neither how to read, nor how to write (forget about solving problems itself), but are always eager to talk about ethical issues to people who do (my humble opinion, of course :-) ).  
A: Littlewood's Miscellany includes a quote from somebody's paper saying 

Part 2 [of the theorem], which is trivial, is due to Hardy and Littlewood

which met the requirements of both completeness and acknowledging prior publication of something H&L had earlier also included for completeness, while coming across as vaguely demeaning  
Your aim should be for usefulness and prior attribution of the result without sounding negative
A: As a reader I would be happy to read a self contained paper with a unified notation and that I don't have to look up a very old paper. 
On the other hand, If I had to review a paper where a significant part of the paper is an old proof in a new language I would check very carefully whether the contributions of the authors are significant enough. Of course if they are,  I would be very happy that they included the old proof.
A: I don't think academic math comes up with general policies for things like this. The dreaded "common sense" should be applied. How well known is the old result, for example? Some things are very old but everyone knows them. 
Or..to give an extreme example: Suppose your result was little more than a corollary of an old obscure result. Writing up the old result in new language and then tacking your result on the end wouldn't look too good... So in such a case I would not include the proof of the old result and just accept that I had a very short paper.
The other extreme is that the old obscure result is fairly short and not too hard anyway, in which case people would believe you/be able to read the old paper if they really had to... so again I would not include the proof of the old result.
If you judge to be somewhere in between: that the old result is very relevant and is hard or subtle in some way but that doesn't overshadow your own work, then people will probably just be pleased to see a nice account of it.
