How to remove activation lock without proof of purchase?

Did you buy the device second-hand (used)? If a used iOS device has been 'Activation locked' then only the previous owner can unlock it, either by providing you with the account ID and password, or by carrying out the steps detailed in How to remove Activation Lock - Apple Support (as he should have done before selling it).

If you are unable to contact him to do this then I'm afraid you will not be able to use the device - there is no other way of unlocking it at all. Apple cannot unlock it for you, nor find the previous owner for you, and neither can the websites that offer to unlock it for a fee: these are scams.

You should if possible return it to wherever you bought it and ask for a refund as in this event the device is completely useless.

If on the other hand you bought it new, and the ID it's asking for is yours but you've forgotten it, go to https://iforgot.apple.com to retrieve the ID and reset the password. See also If you forgot your Apple ID - Apple Support

Apple will not remove the lock if you don't have proof of purchase.

No way to bypass activation lock in such a situation, sorry, unless you can produce appropriate documentation that states you are the original owner.

Regards

Giulio