TCASII RA AT FL410 WHEN AN ACFT ABOVE HAS ENG FLAMEOUT AND DSNDS THROUGH RPTING ACFT'S ALT PRIOR TO NOTIFYING ARTCC OR BEING OBSERVED BY ARTCC.

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THE FLT WAS PROCEEDING WBOUND. WE HAD ASKED FOR 'WRONG DIRECTION' AT FL410 TO GET ABOVE TURB. THE FO AND I WERE IN THE MIDST OF A DISCUSSION RELATING TO OPERATIONAL MATTERS. ZLC ADVISED US OF TFC (AT 12 O'CLOCK POS I BELIEVE) AND ORDERED A 30 DEG TURN TO THE R. MY FIRST THOUGHT WAS THAT THIS WAS A NORMAL SIT. IT IS NOT UNUSUAL FOR AN ACFT CRUISING AT AN ALT NOT NORMAL FOR THEIR DIRECTION OF FLT TO BE VECTORED OUT OF THE WAY OF AN ONCOMING ACFT AT THE APPROPRIATE ALT. (FROM THE BEGINNING WE BOTH THOUGHT THE OTHER ACFT WAS ALSO AT FL410; IN CRUISE.) IT WAS AT THIS POINT THE TCASII ISSUED A TFC ALERT. IT DISPLAYED AN AMBER TARGET BTWN THE 2 1/2 AND 5 MI RINGS (APPROX 3-4 MI) AT '11 O'CLOCK POS; AND 02 FOR THE RELATIVE ALT. WE SEARCHED FOR THE TARGET VISUALLY; BUT DID NOT SEE IT. THE CTLR WAS TALKING TO ANOTHER ACFT; THE ANXIETY LEVEL WAS INCREASING AND THEN THE TCASII COMMANDED 'CLB; CLB NOW.' THE DISPLAY SHOWED A RED TARGET SUPERIMPOSED ON OUR ACFT SYMBOL WITH '00' FOR THE RELATIVE ALT. I IMMEDIATELY PULLED BACK ON THE YOKE; DISCONNECTED THE AUTOTHROTTLES AND INCREASED PWR. AFTER PASSING 41200 FT TO 41300 FT THE 'CLR OF CONFLICT' MESSAGE WAS SOUNDED BY THE TCASII. I ESTIMATE THE ELAPSED TIME FROM THE CTR CTLR'S FIRST ALERT/COMMAND TO THE 'CLR OF CONFLICT' MESSAGE WAS ABOUT 15 SECONDS. EVERYTHING HAPPENED VERY FAST. WE CLBED AROUND 700 FT ABOVE OUR ASSIGNED ALT OF 41000 FT. JUST AFTER THE TCASII SOUNDED 'CLR OF CONFLICT' WE HEARD A LEARJET RPT AN ENG FLAMEOUT AND STATING THEY WERE DSNDING TO 15000 FT TO ATTEMPT A RESTART. THE COPLT AND I BOTH THOUGHT WE MUST HAVE JUST MISSED HITTING HIM. UPON LNDG I CONTACTED ZLC AND SPOKE WITH MR XYZ. HE STATED THAT THE LEARJET HAD BEEN EBOUND AT FL430 AND EXPERIENCED AN ENG FLAMEOUT WHICH RESULTED IN HIM DRIFTING DOWN THROUGH OUR ALT. I BELIEVE HE STATED THAT THE CTLR'S FIRST INDICATION OF A PENDING CONFLICT WAS WHEN HE REALIZED THE LEARJET WAS NOT AT HIS ASSIGNED ALT AND CLOSING ON US. I TOLD HIM WHAT OUR TCASII INDICATIONS WERE. HE STATED THAT THEIR TAPES INDICATED OUR CLOSEST POINT OF APCH WAS 3 1/2 MI AND 400 FT.