Mr Clark- can you explain to me why the jurors only had to prove one piece?
The jury was told it had to find that the falsification of the business records was intended to achieve an illegal goal, and it was presented with several illegal goals that would qualify including violation of federal election laws and violations of tax laws. And jurors were told that so long as they all agreed that the falsification was done for one of these purposes, they didn’t have to all agree on which one. On appeal Trump will argue that this violates the requirement of the unanimous jury or that the charge was illegally vague. But New York law clearly permits this instruction. So Trump may have the challenge of arguing that the NY law itself is unconstitutional.
I'm not sure I understand your question (the jurors don't need to prove anything). I believe you're asking why the jury was instructed that it only needed to find that the falsification was was done for one of three illegal purposes without specifying which of the three.
The quote in your post appears to be correct. As I understand it, under NY law the prosecution is allowed to provide the jury with alternative illegal purposes, and the jury can decide that the prosecution proved at least one of the three. In theory, there could be three different groups of jurors, each group finding that the prosecution only proved different one of the three.
Given that NY law specifies that this is okay, Trump would need to persuade a court that the NY law is unconstitutional, at least when applied in this way.