If we already test each batch of our finished product for the absence of Staphylococcus aureus and Pseudomonas aeruginosa, is it required to test it also for the purified water?

Yes, you are required to test the purified water for the absence of Staphylococcus aureus and
Pseudomonas aeruginosa. It is the general expectation that raw material testing support finished product
testing.