No I would not tell the full time employer anything about it. My only concern would be what happens if my full time job ask me to stay late, work additional hours etc... that interfere with my part time job. So I would let the part time job know.
I'm with DDII & Tensor on this one. Don't tell the Full Time, but let the Part Time know.
When I was in real estate, because it's so cyclic, I had a part time job as a Sales Associate for Dell Computer at Carolina Place Mall. I told them mright up front that I was a real estate agent. They said they were happy to work around my schedule.