I recently started studying advanced mathematics on my own, after taking two classes in college (discrete math and advanced mathematical logic) that spurred my interest in the topic. However, I am struggling a little bit on things that I assume others find easy. I have started a real analysis book (Rudin), and find some of the topics less intuitive and more (seemingly to my brain) arbitrary when compared to the stuff I learned in logic and discrete math.
I specifically feel like I have issues retaining things that I have proved, even if I understand the steps of the proof, because they have no clear intuition in my mind. To give a simple example, consider the following theorem for a field:
Consider $x,y,z \in F$
If $x+y=x+z$, then $y=z$.
The proof for this was fairly easy to me, but I feel like I don't have any intuition on how to remember this. Obviously, this statement is true for the rational numbers, which is a field. But how do I intuit that this is true for all fields? I understand the proof, but don't intuitively understand why its the case by necessity for any field. To me, a field just seems like an arbitrary construct with no real intuition behind it. I feel like I ultimately just have to memorize the result, because without reproving it, I would have no idea whether it was necessarily true or not.