You ask how can you prove that you were laid off. If your request goes to a hearing, you can prove it in at least two ways. First, you can provide your own testimony at the hearing. Better evidence would be an affidavit by the doctor who lost his contract and told you that your job was eliminated. Hearsay evidence, such as an affidavit from the doctor, is admissible at such an administrative hearing for what it is worth (see, Labor Law section 622: "The appeal board and the referees, in hearings and appeals under any provision of this article, shall not be bound by common law or statutory rules of evidence or by technical or formal rules of procedure but may conduct the hearings and appeals in such manner as to ascertain the substantial rights of the parties.") Good luck.
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The outcome will be fact specific. The principal question in a UI appeal is whether the claimant is unemployed through no fault of his/her own. The first issue is the initial cause of unemployment; whether you were laid off because your employment was discontinued, or, less likely, whether you opted to stop work. The second and perhaps more important issue is why you are without work at the present. If you planned to go back to school, and you did return to school, then you are no longer "able and available" for work. In other words, you have chosen to withdraw from the work force, to go to school, and therefore you are not employed because of your own action, and not because the practice laid you off.
The correct answer will depend on the law as it is applied in your State. The outcome on the same facts may vary from State to State. Consult a local attorney.