(Creating this writeup was a mistake, but I can't nuke it so why not try to make the best of it.)

The basic observation is that completion sets (see the statement of the theorem for details) correspond to the states of a deterministic finite state automaton. Reading some string w brings the automaton into that state; subsequently reading a string in its completion set S_{L}(w) brings it into an accepting state, if L is the language recognised by the dfa.