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I need to find a way to make the point that by refusing to contribute what a court would say is his share, H is still expecting me to take care of him.
breton, your best bet is to let the mediator tell him the score. It isn't that you are taking care of him, but rather than he is not taking care of his children. That is why you have a mediator.

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What do I do if H walks out?
Let him go. You really can't stop him. Ask mediator what to do (mediator really can't take sides though). I wouldn't be surprised if the mediator would say get a lawyer. Get in touch with DB coach.

One step at a time.

IMP