right time for the T

[Glenn Lampman]

Situation: completed the 1st half, go to the table, switch the arrow, check with books, they say all OK, go to dressing room. Back on floor 3 min before end of halftime. Buzzer sounds and they wave us to the table and inform us that A8 had played during the first half and they had to add her name during the 1st quarter when she first entered the game. her name had been added after the R had noted the lineup 10 min prior. The R said it was too late to give a T. After discussion we started the 2nd half with a T for entering name after game started. The book (NF) states it is penalized when it occurs?? What is your opinion on giving the T when it became known to the officials, or were we too late to give the T??