In 2013 I installed a BITCOIN CORE (V0.10.2) wallet in my netbook computer and purchased 1 bitcoin. Recently I have wanted to sell but found the wallet out of sync and needing to download the entire Blockchain because the computer crashed years ago due to dud battery. when I get past the date when I purchased my bitcoine it appears in my wallet OK but trying to complete the block chain download takes so long that the netbook Locks up before it is complete - why i don't know, but anyway.................... Is there a way to transfer my wallet or its contents to a new BITCOIN CORE wallet (v0.15) that I have in a much newer computer without completing the blockchain in my old netbook? Put another way - what constitutes the identity of my old wallet? - and can that identity be transfered to my new wallet so my bitcoin appears there?
Bitcoin Core 0.10 was not released until 2015. It is impossible for you to have used that version in 2013 as it did not exist.
However the wallet format itself has not changed since at least 2012 (and earlier really) so it is completely fine to upgrade your software. The wallet.dat file can still be used in Bitcoin Core 0.15. All you need to do is make sure that Bitcoin Core is shut down, go to the Bitcoin Core data directory, and copy the wallet.dat file from there to the Bitcoin Core data directory on your other machine.
I am not sure on the changes from version 10 to current. But you may try to copy your old wallet.dat file in to a separate folder on the new computer (do NOT overwrite), and start with the "-wallet" option as command line parameter. This would specify which wallet to load rather than having to copy or overwrite files (thx to comment from MeshCollider @ Nov 1 at 18:41).