DON'T use the stiff brush at all. Throw it away. Use the soft brush in a back to front motion. It's all you need. I've used this cleaner for years with never problem.
I use the soft brush (built into cap) to drip a couple of drops of fluid to the stiff brush, never touch the soft brush to the stiff brush. That way I don't contaminate the fluid.
I use the stiff brush to clean the stylus. Just be gentle, and only brush back to front. I clean the stylus after every session. That way I know I am starting out with a clean stylus the next day.
I agree with Steve and, in a typically anal fashion, go a step further. I use two bottles of #9. With the first I apply some #9 directly to the stylus tip using the soft brush and then apply a drop to a LAST stylus brush and give the back to front treatment. Then, I apply a couple-o-drops to a second LAST brush and repeat the back to front. Ive used some styli for hundreds of hours and viewing them under the microscope shows them to be clean as a whistle. Of course, using RRL record cleaning fluids helps a great deal in that it leaves no residue (unlike many other cleaners) and the stylus does not get as dirty in the first place.
All that said, it pays top check with the cartridge maker to make sure you don't have a hollow cantilever that might wick fluid up into the cartridge - though I've never had a problem with that or stylus adhesion with any cartridge in about 15 years of use.
There are quite a few alternatives out there these days, but I prefer the tried and true - at least for me. I contacted Musical Surroundings awhile back regarding an unrelated matter, and found very interesting their subsequent comments regarding experience with stylus cleaners and damaged/returned cartridges. It is worth a discussion with them if you have the time and interest.