Droid - I assume that the '12' in the name of the girls's bike denote 12-speed, i.e. 2x6.

If both bikes use a 6-spd Shimano cassette, you should have no problem swapping the cassette onto your freehub. Some 6-spd setups use freewheels though (cassette and freehub mechanism are one unit which screws onto threaded hub body), and you can't interchange a cassette onto a freewheel-type hub or vice-versa.

If they are both of the same type, there's no harm in taking the cassettes/freewheels off and giving it a go. Unless the lowest ratio on your intended new set is massive (some Shimano cassettes go to 30-odd teeth) you should have no big derailleur compatibility issues.

See the sticky post of shortcuts to Sheldon Brown's website at the top of this forum for links to specific technical aspects of this.