A DC9 FLC OBSERVED A BEECH 1900 XING THE DEP RWY ON TKOF FROM TVC.

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ACFT X TAXIED TO RWY 28 VIA TXWY D ON 121.8. STOPPED SHORT OF RWY TO WAIT FOR WTS -- MONITORED TWR FREQ ON 124.2. CALLED READY FOR TKOF. TWR CLRED FLT TO POS AND HOLD; RWY 28. WHILE MOVING ACFT TO RWY 28; TWR CLRED ACFT X FOR TKOF RWY 28. FO ACKNOWLEDGED BOTH CLRNCS. CAPT TAXIED SLOWLY ONTO RWY; STAYING CLOSE TO RWY THRESHOLD TO TAKE ADVANTAGE OF FULL LENGTH; AND RELEASE CTL OF ACFT TO FO. FO TOOK CTL OF ACFT AND APPLIED TKOF THRUST. AS FO APPLIED PWR; SCANNED RWY AHEAD AND SAW BE1900 CLBING THROUGH RWY 28/36 INTXN. DID NOT HEAR TWR OR GND TALK TO BE1900 PRIOR TO INCIDENT. SUPPLEMENTAL INFO FROM ACN 450508: UNKNOWN TO US THE OTHER ACFT MISTOOK OUR TKOF CLRNC AND TOOK OFF. MY TKOF SCAN WENT TO MOSTLY OUTSIDE AFTER OUT 80 KT CALL AND BY THEN THE OTHER AIRPLANE WAS WELL CLR AND N OF THE INTXN. NO REAL CONFLICT OR CLOSE CALL THIS TIME BECAUSE WE TOOK SO LONG TO ACTUALLY ROLL ON OUR TKOF -- MIGHT HAVE BEEN MUCH WORSE. SUPPLEMENTAL INFO FROM ACN 450404: WE THEN RECEIVED 'ACFT Y YOU ARE CLR FOR TKOF RWY HEADING AND 3000 FT.' MY CAPT REPEATED THIS BACK AND SAID 'WE NEED A SECOND.' THE CTLR SAID 'ROGER.' WE THEN TURNED ONTO RWY 36 AND COMPLETED OUR TKOF FINAL ITEMS CHKLIST AND THE CAPT SAID TO ME 'CLRED FOR TKOF; RWY HEADING AND 3000 FT.'