Thanks for the additional information.
If the agreement states that your customer is obliged to serve 3 months notice before termination, then it will usually be bound by the remaining contractual obligations during the 3 month notice period.
The next thing to consider is precisely what obligations they have and what rights you are entitled to. For instance, if they agreed to use you as their exclusive supplier of H&S courses, they may not be able to use an alternative provider until the end of the three months. Also, if they agreed to purchase a minimum amount of services from you on a monthly basis, they may also still be bound.
If it is completely "ad hoc", there are no minimum orders or exclusivity requirements and they are not obliged to make any payments to you unless they choose to use your service, then they may well decide to start with their new provider before your contract terminates. Of course, depending on the terms of your agreement, you are unlikely to be obliged to assist them with their early transition if they do this.
Check the contract. Even if they won't be in breach if they start with the new provider before the end of the termination period and there is no economic disadvantage to them in doing so, there may be something that at least makes it adminstratively or practically burdensome for them. This may then provide you with some leverage.
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