When you’re a fireman whose job is to plan which vehicles go where or when you need to precisely specify which type of fire vehicle. Non-firemen usually say Feuerwehrfahrzeug or even Feuerwehrauto.

OK, you got it then, I believe. P3 is specifically built so that P3(xn)=yn for n from 1 to 4. The proof lies in its construction. I guess the sentence can be understood as “we know it works because we built it like that, however you may verify it yourself”

Replace the words “Convince yourself” with “You can verify” and it might make more sense.