New Zealand Law
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If the person who sold to the property had taken the gate from the neighbour, then you never had good title to the gate. You did however buy the equipment in good faith, and like you, I am inclined to suspect the neighbour has a story which is too good. If the neighbour was right, then they could have claimed the gate back, but could not enter your property without tresspassing.
Your neighbour can explain what the gate looked like, because he had claimed that he entered the property and he has seen the gate very recently. I think you should take this matter further with the police, because even if the gate was stolen, your neighbour was wrong to enter your property without permission to retrieve his property.
Your reaction was to go above the police officer to his sergeant, and you should ask why they are protecting someone who wrongfully entered your property. I doubt very much if he could have identified the gate without seeing it recently, and normall the police would not intervene in matters they see as civil claims.
if the gate was of some value, and it actually was stolen, then you can recover the value from the person who sold you the property, because he did not have good title to the gate. that may not be a very effective remedy after this time if you dont know where he is now.
Actually the police officer is wrong about the entry. it is trespass unless he has permission, and you dont need a notice. But you can bring a claim in the Disoutes Tribunal for the gate and for the damage and the trespass. You can file this at your local court or on line at this address http://www.justice.govt.nz/tribunals/disputes-tribunal/apply-online