I'm assuming R is a recycled material that gets reprocessed from product stream P1.
Given the fullness of time and the complete reprocessing of R so that at the end of production, no R is present then you can say that F1 + F2 + F3 = P1 + P2 + P3.
However, processes and processing plants rarely operate that way. Usually there will be an amount of R within the system. To account for this F1 + F2 + F3 = P1 + P2 + P3 + R.
With material balance, conservation of mass is paramount - what goes in must come out.
Regarding your concerns about F3 being added after P1 and P2 have been removed, F3 is added to the residual from the P1 and P2 process to produce P3. Your concern would be valid if there was no residual feeding into the second process to which F3 is added.
You need to consider the system as whole when it comes to material balance. There are three inputs/feeds and three products, with some of the feed recirculating.