The same way you prove anything else in a legal case, by sufficient evidence to convince the trier of fact that it is sufficiently likely to be true ("sufficiently likely" depending on the applicable standard of proof, which varies based on the context which creates the need for proof.)
Legal proof (even to criminal conviction standard of proof beyond a reasonable doubt, and even moreso to the standards used for civil liability and other purposes) isn't logical proof.
The same way you prove anything else in a legal case, by sufficient evidence to convince the trier of fact that it is sufficiently likely to be true ("sufficiently likely" depending on the applicable standard of proof, which varies based on the context which creates the need for proof.)
Legal proof (even to criminal conviction standard of proof beyond a reasonable doubt, and even moreso to the standards used for civil liability and other purposes) isn't logical proof.