Mafia 2 Definitive Edition: Game won't install. Why?

I bought Mafia 2: Defintive Edition for the Ps4 and have the problem that I can't play the game after the first chapter (i.e. Where Vito fights as a soldier in Italy) because the game is always displayed to me can't continue until it is fully installed. However, it is installed and I can't find any indication anywhere that it is still installed. I had the game reinstalled before, but then I have another problem after the first mission / chapter.

Are you sure that the complete game has already been downloaded? Look under your downloads.

Just play the first mission and wait for 15 minutes… That's it… No need to be online… I have mafia trilogy pack on disc