If your property rights are being challenged you need a attorney to review what authority your neighbor has over you. However if you are on his property he has to get a court order to remove your property, likewise you need to be able to claim adverse possession to keep it and 7 years is not really enough in most cases. Realistically the cost of moving a shed is miniscule compared to a lawsuit.
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The notice requirements, if any, would be found in the zoning ordinance. There might be no notice required, if a Boarding House is a permitted use in the LR2 zone. Or it might have required a rezone, which would require some notice, typically publication in a newspaper. Or it might be a conditional use in the LR2, which typically would require notice in writing to adjacent property owners.
Either look at the city zoning ordinance, or contact a land use attorney in your area.
By the way, I agree with the other answer on moving the shed. And they are not required to get a survey, but they won't get a court order to require you to move a building without one.
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Refer to your city's code of ordinances, and look for notice requirements for new development. A "land use/zoning" attorney in Seattle would be able to help you determine this answer quickly. You might be able to find the answer by simply calling the city planning staff and talking with them about the situation.
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