It definitely takes a second to develop that intuition, I remember being quite confused as well. As I got more into CS, I began to think of proofs similar to ho

It definitely takes a second to develop that intuition, I remember being quite confused as well. As I got more into CS, I began to think of proofs similar to how I do unit tests: you externalize or “fix” as many as the inputs as you possibly can, and then abuse the remaining degrees of freedom as much as possible to show that your solution is complete.