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Note that you will only get a license number immediatly after purchase, not another download.
The application is the same as the demo!

To get your demo to the full version, enter the license number into the "registration" window. (file->register)

As the license number is case sensitive and the application will quit if you make a mistake, it's best to use copy/paste to enter it in the "registration" window.

As most of the programs undergo constant changes, it's very well possible that every now and then there will be a newer version. You can install that and use the same number, unless the version number goes up by a whole point (e.g. 1.8.2 to 2.0)