Within a few hours a user had suggested it could well be Coenosia tigrina, one of a group of killer flies. Putting the name in Google and looking at the photos it shows I am very sure this is the fly I saw and it certainly looked as though it was attacking the other fly. Some information about the killer fly can be found on BugGuide.
The iSpotNature site looks to be an excellent resource for finding the identity of many unknowns in the natural world. What is the iSpot site?.... This is from their help page: