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Yes, but if you see an app in the app store and you can't remember if you previously bought it or not, you can't tell until you click the "buy" button. The "purchases" list in iBooks lets you see all the books you've bought before you redownload them, so no more guessing.

My apps show "installed" in the app store if I've bought them previously.

As to buying apps for 2 Ipads, i have 2 (one bought for my husband) and for those apps that he wants that I have bought he simply chooses them and since his Ipad is registered with my account it asks if he wants to redownload them for free. I haven't bought any books in Ibooks--just Kindle, Nook (Barnes & Noble) and Stanza. We use the same account for both Ipads. .
 
My apps show "installed" in the app store if I've bought them previously.

As to buying apps for 2 Ipads, i have 2 (one bought for my husband) and for those apps that he wants that I have bought he simply chooses them and since his Ipad is registered with my account it asks if he wants to redownload them for free. I haven't bought any books in Ibooks--just Kindle, Nook (Barnes & Noble) and Stanza. We use the same account for both Ipads. .

They show "Installed" if they exist on your iPad. If you have bought an app and deleted it - it will not show you anything other than "Buy Now" which I think is stupid of Apple - they should somehow indicate what you have purchased before in that same window.
 
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