Read the following.
The standard of civil proceeding is proponderance of the evidence (more likely true than not, 51% likely to be true). If they lie and you said they lie, the evidence is 1:1 and it is 50% < 51%. For every of their lie you need to have two piece of evidence against their lie. If not, their lie washes out your statement and the court cannot make a ruling based on that piece of fact.
Since you cannot enter new evidence like "In several states" below, you need to review your existing evidence based on the "proponderance of the evidence" standard to see if you have any solid evidence to win the case on appeal.
https://www.nolo.com/legal-encyclopedia/free-books/small-claims-book/chapter23-4.html
In several states, either party can appeal and have the case heard over from scratch. In other states, only the defendant can appeal (in most cases), but if the defendant does file an appeal, the whole case is presented again by both sides as if the first hearing hadn't occurred.
When a whole new hearing is allowed on appeal (it's called a trial de novo), you simply argue the case over, presenting all necessary witnesses, documents, and testimony. Starting from scratch is required because records often aren't kept at small claims court hearings. However, in some courts, judges record hearings, and these recordings are available to the judge who considers the appeal as part of the reargument of the case.
Questions of Law
In many states, appeals can be based only on questions of law, not on the facts of the case. What's the difference between law and facts? It's best illustrated with a couple of examples.
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