Here is the proof.
Here are very simple 3 steps, easy to comprehend. Once understood, you have solved one of the biggest puzzles of your life.
(1) Just think about any moment in time.
At that particular moment, it is 'present'.
This 'present' is already here. You can't change it now. Hence 'present' is UNCHANGEABLE.
(2) Before moving to next moment, consider another factor 'Laws of Nature'.
Sequence of events in this universe always follow 'Laws of nature' (agreed that not fully known to us, but no doubt these laws must be UNCHANGEABLE). Progression of universe to next moment in timeline must follow these laws.
(3) Ok, here are we.
Next moment is result of status of universe in last moment ('present') and rules to calculate next moment ('Laws of Nature') under which time will progress to next moment.Now very easy to understand. First moment called 'present' in (1) is UNCHANGEABLE. 'Laws of nature' in (2) are UNCHANGEABLE. Hence next moment i.e. result of this equation in (3) must be UNCHANGEABLE. No doubt there is only one possible outcome to happen in this next moment.
This scenario is true for any given 2 moments in time. Yes, your fate is sealed. Nothing is random or choice or free will.