It follows from them quite clearly that a pair of unique, integral legs must have a unique integral hypotenuse; i.e. this hypotenuse cannot give a rectangular integral triangle with any other pair of integral legs.

As was pointed out to me by several worthy parties on sci.math this week, it is indeed possible for more than one distinct triangle to share a hypotenuse.  For example, 

33² + 56²  =  39² + 52²  =  25² + 60²  =  65²

The name "Nafets Azereb" is rather odd, to say the least, and "Nafets" is just "Stefan" backwards. I believe he posted the proof as a deliberate joke.