Ilki,
You say, "The prepositional complement in the subordinate clause is raised out of its original place and placed into the subject-slot of the matrix clause via a process called "raising"."
That is probably a good description of what is going on here, but once again there appears that difference between the British and American systems of analyzing (analysing) sentences.
Within the American system we call "prepositional complements" "objects of prepositions", we call the "matrix clause" the "main (or independent) clause" and we do not recognize "raising".
I believe that the British system is probably more precise, but it does not lend itself so well to R-K diagramming, which is a very fun activity.
The Parser's original question was, I believe, how would one diagram the sentence. Perhaps you could show that. I would like to see it.
In about a week, I will be teaching a complete course on Reed-Kellogg diagramming. I will be doing this online at Youtube channel "mrbisse1".
You might be interested. It shouldn't take more than about two weeks.
Frank