ANSWER: You don't have to prove it, in the sense of being required to produce objective medical evidence. In certain clear-cut cases (i.e. that of the typist whose hand was cut off), it is easy to prove disability. But such cases are the exceptions. More frequently, disabling symptoms are impossible to prove, and, contrary to popular belief, the law does not require you to do the impossible. Subjective but very disabling symptoms (such as fatigue, chronic pain, phobias, depression, and allergies) are usually impossible to prove, because no objective medical evidence is usually available. Yet they are among the most common disabilities for which government employees seek and obtain a monthly disability annuity. Employees with subjective symptoms, regardless of their medical condition, face the greatest difficulties before OPM.
Sometimes, even when you can provide objective medical evidence to support your claim, OPM may not be interested in looking at it. In one case, I sent a client's x-rays to OPM to demonstrate that he had a slipped disc. OPM refused to accept them, saying that it had no place to store x-rays. You would think that if only to prevent potential fraud, OPM would have demanded such information, not blindly turned it away.