You can attempt to show intent to a judge and jury with physical evidence and witness testimony. It just needs to be a better story with better props than the other side can put together!
During the discovery phase of a legal case the court will order inspections, evidence to be produced, depositions, expert reports, etc. Somewhere in there is an arrangement of facts that signifies some kind of intent behind an action.
Ok well yes you can 'prove intent' beyond reasonable doubt - I perhaps wrongly (given the context) took other commenter literally. You can't ever actually prove what a person intended, if they haven't claimed or claim otherwise themselves. But yes you can of course satisfy a judge or jury of peers that something was overwhelmingly likely their intent, and they are just lying about (or not admitting to) it.
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.
(.. Short of finding a note that 'I OJFord declare that I intend to [do something I'm later accused of] signed [etc.]' I mean .)