I see, you did miss an important part of this letter, above the red one titled "You do not keep your line at this address”. See below. I'm guessing the letter above has been translated with Google translate. This part specifically says that you need to let us know that you want to cancel your contract with us, if you haven't done this already. Another provider taking over the line does not mean your contract is being terminated. Legally speaking: you are the only one who can terminate the contract. 

The question is if you used the Switch service (Overstapservice) offered by your new provider. If so, you will need to contact them to straighten this out. In that case they are responsible for the switch without overlap of cotst. 

Can you confirm if my contract is now cancelled with effect from 13th Oct as was told to me in one of the call with customer care. Also what can I do to get this double invoice paid..Can it be paid back by KPN??

Yes, I can confirm that your contract has been cancelled on the 13th of October. The only thing you can do is check your confirmation of your new contract to see if you've used the 'Overstapservice’, meaning that your new provider is responsible for the cancellation of your old provider (KPN) without overlap of costs/invoices. If so, then they should fix this. 

Thanks for the response Alex. Now google translation needs to be blamed however for information, before making a switch I did ask the process at KPN shop in Eindhoven centrum and was told that it would be done by new service provider and i need not do any thing. These it was in English only(without translation), Because of gaps in communication between two providers I as a customer need to suffer and no one is willing to provide a solution,

This is not about willingness but responsibility. If you used the overstapservice your new provider is responsible for the transfer without overlap in monthly fees. If you used this service and you have the confirmation form the new provider, then the new provider must fix this. Simple rules. If it was the other way around then we would be responsible. I hope you can work something out with them.