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I uploaded a picture of my rear wheel to make things easier
That's a freehub for a cassette, rather than a screw-on freewheel.
From what's been said above, you'll struggle to find a 7 speed cassette.
However, according to VB your shifters can be run either indexed or friction, so you could still use them with your 6 speed cassette.
Scilly.Suffolk
The VeloBase page for the SL-6400 shifters say they are indexed (http://velobase.com/ViewComponent.aspx?ID=75c91ac5-4408-49e5-a9f1-d6581e1be9d5&Enum=104) I was under the impression that this meant they were not friction shifters but would be happy to be told I'm wrong :)
EDIT: I uploaded a picture of my rear wheel to make things easier