ACFT EQUIP PROB. ENG OVERHEAT; FIRE LOOP FAULT; FIRE LIGHT AND WARNING BELL WHILE IN CRUISE. FLC COMPLETED CHKLISTS; SHUT DOWN THE ENG AND FIRED EXTINGUISHER BOTTLES. EMER DECLARED AND FLT DIVERTED TO LAS. POST LNDG INSPECTION FOUND SHORT IN FIRE DETECTION CIRCUIT.

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DURING CRUISE WE GOT AN ENG OVERHEAT LIGHT ON #1 ENG. AFTER THE FO COMPLETED THE CHKLIST THE LIGHT WENT OUT. AFTER SEVERAL MINS THE LIGHT CAME BACK ON; FOLLOWED SHORTLY BY A FIRE LOOP FAULT LIGHT. WE GOT CLRNC TO DIVERT TO LAS WHILE COMPLETING ENG OVERHEAT CHKLIST AND FAULT LIGHT CHKLIST. BEFORE BOTH CHKLISTS WERE FINISHED WE GOT A FIRE LIGHT AND BELL ON #1 ENG. WE DECLARED AN EMER; COMPLETED FIRE/SEVERE DAMAGE CHKLIST. WE FIRED BOTH EXTINGUISHER BOTTLES AS LIGHT WOULD NOT GO OUT. WE LANDED IN LAS WITHOUT FURTHER PROBS. FIRE TRUCKS RPTED NO EVIDENCE OF SMOKE OR FIRE AND WE TAXIED TO THE GATE WHERE MECHS CONFIRMED A SHORT IN THE #1 ENG FIRE DETECTION SYS. FROM START OF PROBS TO LNDG WAS LESS THAN 15 MINS. AFTER OVERHEAT LIGHT FIRST CAME AND WENT OUT; I HAD COPLT FLY WHILE I GOT A HOLD OF COMPANY. WHILE I WAS TALKING TO THEM; FURTHER PROBS DEVELOPED; SO I DIRECTED HIM TO CONTINUE TO FLY TO LAS. WE HAD A JUMPSEAT PLT AND I DIRECTED HIM TO MAKE A PA TO THE PAX ADVISING OF THE DIVERSION; AFTER WHICH HE HANDLED ATC COMS. I COMPLETED CHKLISTS KEPT AN EYE ON ACFT POS AND TALKED TO THE 'A' FLT ATTENDANT. AFTER THE SECOND BOTTLE HAD BEEN FIRED; THE FLT ATTENDANTS MENTIONED THEY NEVER SAW ANY SMOKE OR FIRE FROM THE ENG; WHICH FURTHER CONVINCED ME WE HAD A MALFUNCTION OF THE FIRE DETECTION SYS; SINCE THE FIRE LIGHT STAYED ON THE ENTIRE TIME FROM INITIAL BELL TO THE GATE. CRM TRAINING CAN TAKE FULL CREDIT FOR THE UNEVENTFUL COMPLETION OF THIS FLT. WITHOUT THE HELP OF ALL 3 PLTS AND 3 FLT ATTENDANTS WORKING AS A TEAM WE WOULD HAVE HAD TO DELAY LNDG TO COMPLETE CHKLISTS AND COMS OR LNDG WITHOUT CHKLISTS BEING COMPLETED.