Actually, the airlines DO NOT give TA's commission. In fact, we are charged up to $39 per ticket to make the reservation for you. (charged by the booking system) If you are not being charged by your TA for making those airline reservations it is because they want to do a great service for you. They are paying $39 out of pocket to keep you happy. You might be paying the fee without knowing it, because your agent might just be quoting you the Centrav price that includes the fee. I waive the Centrav fee when I am booking for a good client because I want them feel well taken care of.
In answer to the original question, yes, the first TA would lose the commission. My personal opinion is that it is incredibly unethical and frankly crappy behavior to try and steal a booking from another TA. We don't actually get paid giant commissions. After overheads like insurance, paying for our website, rent, a phone line, gifts for our travelers etc. last year I netted 5% of my sales. If someone convinced one of my travelers to book with them I would be gutted. Not just because I would lose the commission I was counting on, but because I work hard for my travelers and I want them to be happy.
This whole thread has been kinda rough on TA's. We love planning travel for you and we are passionate about you having the time of your lives. Our success literally depends on you being thrilled with your experience so we go out of our way to make sure you are having fun and not worrying about the details.