There is a comment here, saying that many hard theorems require one to build a complex branch of math and use it to prove a single statement. So I asked myself: what does it really look like to prove a theorem at such level?
I can tell what's going on in a programmer's mind. Software is very much like an imaginary mechanism and software engineers are mechanics. For example, this site is a database connected with a html page, so a programmer literally imagines a big gray building that means the database, another building that means the html page and a pipe that connects them. The database has a few tables: one for user accounts, another for posts like this, another for comments. So a programmer imagines 3 big blocks inside that building that are connected with pipes transferring data. Next to the database there is a controller device that sends and receives messages in the pipe connecting it to the users. This analogy continues to tiny things like classes, methods and variables. The entire HN forum looks like a big multi-dimensional city-like structure with numerous pipes connecting pieces together. Experienced programmers not only organize this city well, but can also predict and eliminate complexity, e.g. they know that if you put that kind of building over there, others will inevitable connect to it tens of pipes and the entire city will be a mess, where you see a pipe and have no idea what it connects to and what will happen if you cut it.