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To prove a statement false, you need ONE example of when it is not true.To prove it true, you need to show it is ALWAYS true.
A segment need not be a bisector. No theorem can be used to prove something that may not be true!
The same way you prove anything else. You need to be clear on what you have and what you want. You can prove it directly, by contradiction, or by induction. If you have an object which is idempotent (x = xx), you will need to use whatever definitions and theorems which apply to that object, according to what set it belongs to.
They are a set of conditions which need to be fullfilled to prove that a specific organism causes a specific disease.
You can only do it if either the outliers are way out - so far that they must be odd, so far that there can be no argument, no need for statistics to prove them to be outliers, or you need to prove that they are outliers using statistics - something like Grubb's test. To do that, the simplest way is software.