Could I use a word the as a subject, omitting the actual one? Just like this:
The man has bought a hat; the [hat] fits him.
It's a Russian joke, by the way.
The correct interpretation would simply be:
The guy buys a hat and it fits him.
The idea here is that "Take first, think later"… First he buys the hat, then he tries it on and, what do'ya know, it fits him. It is a poor joke in Russian which isn't quite popular, but there is a better joke with a play of words about a guy, a shop that sells hats, and a woman, who is the shop keeper.
And to answer your question - no.