I presume there are no rights of cancellation ?
Is there a written contract at all?
It doesn't matter whether a contract was signed or not. One clearly exists.
The problem with this is that rights of cancellation are not automatic. In fact, they are presumed to not exist unless you negotiate them. A contract is not a contract is either side can cancel.
Therefore, the fact that both of you were silent on the point actually supports their position I'm afraid.
You could try to rely on their duty to mitigate their loss.
The reality is that they can sell their personal trainer to another unless exceptional circumstances arise. If so, then you would be entitled to recover minus any actual costs sustained.
Can I clarify anything for you?
It is not an implicit agreement. It is a contract.
Not of this particular term. It is standard in any contract.
Yes, or otherwise agreed.
Most contracts are not reduced to writing. They remain a contract.