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I keep my 12 word mnemonics carefully off line. But suppose I just had my private key left after some disaster or other could I then, after some procedures, log in to my wallet ? I can see that I can get from Private Key to Public Address by a Python Routine but then I seem to be unable to progress further.

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Yes. The main idea of mnemonic backups is that the mnemonic encodes a master private key. All other addresses that your wallet generates are derived from this initial secret.

To recover your transaction history and funds, you would need to go further than just to derive a single public key. You would need to rederive all of the wallet's addresses and then search the blockchain for any transactions that pertain to these addresses. The simplest way to achieve a recovery would be to import the mnemonic into the same wallet software that created it, or another wallet that implements the same derivation scheme (of which there are multiple, unfortunately).

I'd recommend that you additionally note what software was used to create the mnemonic, (and which wallet software you used to transact from the mnemonic, if the two differ). If you know it, you might also want to note the derivation path with your backup.

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