You have to consider who has the deepest pockets. Who has the most money to be able to risk going to court?
If they simply turned the software off, would that be the end of your business? If they did that, would you then be able to fund the litigation.
At the moment, to an extent, they have you over a barrel until this issue is decided by a court.
You know you don’t have the agreement. What you don’t know is whether they have the agreement or not and they are simply choosing to say they don’t have it because it may be that the agreement that was signed and that you inherited along with the company, did not have any mention of transfer and that is why they don’t want to produce it!
If there is nothing mentioned in the document about transfer, then it is transferable.
Now, this company has presumably been in business for some period of time and presumably, they have other agreements with other companies from the time and it seems unlikely that all of those have been lost or destroyed and hence, it would potentially be possible for them to let you have another from another customer to show the wording. If this gets to court, I would probably raise that in court, rather than now.
You want to keep your company going and at the moment, they are holding a knife at your throat.
There is nothing to stop you therefore either negotiating or if they will simply not negotiate, paying the money under duress/pressure and then taking them to court to have the issue of whether the software is transferable or not determined by the court.
If you think that having the software turned off is a real threat, and would be a real threat to your business, you could apply to court for an injunction preventing them turning off the software in the short term, pending the outcome of a court hearing to have the whole thing determined.
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