Administrative foul clarification?

[jritchie]

2 minutes into game player A6 checks in game, after coming into game and running up and down the floor for couple minutes commits a foul, after reporting foul, scorer now tells us A6 not in book. So per rule 10-1-2, when discovered, we have a T, add players name and number, shoot 2, team B gets ball at division line. Couple minutes later A45 checks in, scorer awake now, realizes that there is no A45 in book, they have A55 instead, so we go and explain we have already had an administrative foul that covers all of the changes listed in 10-1-2, and to go ahead and change it without another penalty. Coach going nuts wanting to shoot 2 more free ones, just wanting to make sure we got this right. Is how I am reading the rule correct? One T for all of the corrections that need to be made, or did I kick it? I know it says when discovered, that is what has me 2nd guessing I think. If you discovered a bunch of numbers wrong, you could change them all with one T, but where these were at different times that is what got me thinking about it.