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Many questions on this site ask some variant of the question: "how does one prove something in court?" Paraphrasing a few:

Generally, how do you prove a fact in issue in litigation?

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The default: present relevant evidence, including testimonial evidence

Outside of special regimes (discussed near the end of this answer), you prove a fact by introducing evidence. The default is that all relevant evidence that bears on a material fact is admissible. Evidence is relevant when, if is were to be accepted, it would make the fact in issue more or less likely to be true. Evidence comes in many forms: testimony of the parties or witnesses, documentary evidence, physical evidence, expert opinion evidence.

If you are wondering, "How do I prove X?" Ask yourself, why do you believe X? Or, how do you know X? Or, how has the event X left its mark on the world? Whatever has led you to believe X probably is the evidence that you would want to introduce to the court to help prove X. Perhaps you saw X: you can tell the court you saw X. Perhaps you took a photo of X: you can present that photo in court. Perhaps you have a receipt for X: you can show that receipt in court. Perhaps X is a proposition about your own actions: you can tell the court about those.

Experts

Some propositions require special expertise to evaluate, so a lay-witness or fact-witness would not be able to testify about them, and a fact-finder (judge or jury) would not be able to readily draw inferences about them. For example: the value of a home, the causation of psychological symptoms after a vehicle accident, or determining the provenance of digital artifacts. To prove propositions like this, one would need to provide expert opinion evidence.

Obtaining the evidence

Evidence can be obtained from the other party during the discovery process. This involves mandatory disclosure and production of documents from the other party, and includes opportunities to examine (question/depose) the opposing party under oath. Evidence can also be obtained from third-parties using subpoenas. These processes are subject to objections relating to relevance or privilege. Expert opinion evidence can be obtained by contracting with an expert.

Not all (potentially) relevant evidence is admissible

However, some evidence will be inadmissible despite it being potentially relevant. I will only present a few categories, some very general and some more specific:

  • hearsay (unless it falls within an exception to the hearsay exclusion),
  • privileged material (unless it is a case-by-case privilege or a discretionary privilege and the person seeking to admit the evidence demonstrates to the judge that it should be admitted),
  • sexual history evidence of a sexual assault complainant when it will be used to support an inference that the complainant is more likely to have consented to the sexual activity that forms the subject-matter of the charge or is less worthy of belief (this is because it has been recognized as not relevant for this purpose),
  • material that was obtained in contravention of the Charter and where the court has decided that the remedy for the Charter violation is exclusion of the evidence (in the United States, there is stricter, exclusionary rule),
  • there are many more.

There are also some meta-rules about the evidence that may be used to impugn a witness's credibility, themselves at trial to provide evidence, but for now I am leaving those out of this fairly summary answer.

Mens rea

It is frequently asked here how the prosecution could prove mens rea (often requiring a showing of the accused's internal knowledge, intention, or purpose).

Mens rea is almost always proved through indirect evidence.

The Supreme Court of Canada has said (R. v. Docherty, [1989] 2 SCR 941):

To the extent that direct evidence of intent is almost always difficult to obtain, the Crown may ask the court, absent any evidence to the contrary, to infer intent from the fact of the conduct.

And (R. v. Boulanger, 2006 SCC 32, at para 57):

As with any offence, the mens rea is inferred from the circumstances. An attempt by the accused to conceal his or her actions may often provide evidence of an improper intent.

For an example, see the model jury instructions for the intent requirement of murder:

To determine whether the Crown has proved that [the accused] had one of the intents required for murder, you must consider all the evidence, including the nature of the harm inflicted, and anything said or done in the circumstances. You may infer, as a matter of common sense, that a person usually knows what the predictable consequences of his or her actions are, and means to bring them about.

Or for harassment (which requires knowledge that the conduct is harassing):

To determine whether [the accused] was aware that his/her conduct harassed [the complainant], you must consider all the evidence including anything said or done in the circumstances.

As with any fact at issue in a case, mens rea will need to be proved through evidence, even if it is circumstantial or indirect. This can include:

  • the accused's testimony;
  • the complainant's testimony (especially about anything the accused said);
  • evidence about what part of the complainant's body the accused was looking at;
  • post-offence conduct;
  • inculpatory statements made to police;
  • etc.

And absent evidence to the contrary, the prosecution can be benefit from the inference that an accused intends the ordinary consequences of their actions.

Special evidentiary presumptions

For some kinds of questions there are special presumptions or forms of evidence that are specified by statute. For example, in Canada and the United States, registering a copyright creates presumptive proof that the copyright exists and is owned by the registrant. In Canada, breathalyzer results are conclusive proof of the blood alcohol concentration if certain conditions are met.

Some of these regimes are not subject to challenge: they are legal facts even if they are not objectively true (e.g. the breathalyser results). Others create presumptions that can be overcome by contrary evidence (e.g. copyright ownership).

Weighing the evidence

The trier of fact (the judge or a jury) then is to weigh all the admissible evidence, including by weighing the witness and party testimony according to its credibility and reliability after testing through cross-examination, to come to a conclusion on the ultimate question(s) at issue.

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    The biggest issue that comes up when people ask this question is a failure to understand that testimony under oath is evidence, because there are colloquial senses of the word "proof" in the English language that excludes an aggrieved person's own statements about what happened.
    – ohwilleke
    Commented Dec 2, 2022 at 5:42
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    Presumably a complete answer to this question for a specific jurisdiction would be a rather large book in multiple volumes. Commented Dec 2, 2022 at 9:39
  • Missing: parties simply agreeing to facts to save costs. E.g. A and B agree that A bought the thing, they dispute if it was defective
    – Trish
    Commented 6 hours ago

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