A fair point. And it's likely you're correct.
Although using the term 'prove' has specific legal meaning that many (myself included) folks would associate with the use of that term.
As I said, there's what you believe and what you can prove. Believing you can prove something may be well founded, but at least in the US nothing is actually "proven" until it has been adjudicated -- and even then contrary decisions (e.g., in an appeal) can "un-prove" stuff.
Perhaps a better phrasing would be "believe they can prove".