UNAUTH TKOF CREATES A POTENTIAL CONFLICT WITH ANOTHER DEPARTING ACFT.

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AT THE END OF AN INSTRUCTION NIGHT WITH A MULTI-ENG STUDENT WE RETURNED TO DPA ARPT AND LANDED ON RWY 10. WE DECIDED TO TAXI BACK FOR 1 MORE TKOF AND LNDG. TFC WAS USING BOTH RWYS 15 AND 10. AN SMA WAS GIVEN A CLRNC TO TAXI INTO POS AND HOLD RWY 15 AND; AT THE SAME TIME; MY STUDENT AND I WERE ALSO GIVEN A CLRNC TO TAXI INTO POS AND HOLD RWY 10. IMMEDIATELY FOLLOWING; THERE WAS A SERIES OF CLRNCS GIVEN AT THE SAME TIME WHICH IS WHERE OUR CONFUSION BEGAN. THE SMA IN POS ON RWY 15 WAS 'CLRED FOR TKOF NO DELAY' IMMEDIATELY FOLLOWED BY A CLRNC FOR ANOTHER SMA Z WAITING AT RWY 15 TO 'TAXI INTO POS AND HOLD FOR TFC DEPARTING RWY 10.' MY STUDENT; BELIEVING THE TKOF CLRNC WAS INTENDED FOR HIM; BEGAN TO ROLL. I DIDN'T QUESTION HIS DECISION AFTER 'TFC DEPARTING RWY 10.' MY STUDENT RECALLS ANSWERING THE TKOF CLRNC WITH OUR CALL SIGN 'ACFT X.' AS WE BEGAN TO ROTATE; I NOTICED THE SMA TFC APPROX 200 FT AGL XING THE DEP END OF RWY 10. NO EVASIVE ACTION WAS NECESSARY; AND WE WERE NEVER CLOSER THAN APPROX 1500 FT FROM THE DEPARTING SMA TFC; BUT THE RESULT WAS LTSS. I BELIEVE THIS INCIDENT WAS CAUSED BY THE CONFUSION IN THE SERIES OF CLRNC AND THE SPD THEY WERE GIVEN. THIS COULD HAVE BEEN PREVENTED BY A FULL READBACK IN CLRNC RATHER THAN JUST OUR CALL SIGN.