ACR ON OCEANIC FLT HAS CLOSE PROX WITH CORPORATE JET.

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FLYING E DIRECT TO DDP (DORADO SAN JUAN) AT FL390 AS APPROVED BY PREVIOUS MIA SECTOR; TCASII AURAL WARNING INDICATED TARGET AT 12 O'CLOCK; 9 MI. WAITED BRIEFLY TO ESTABLISH TRACK WHICH WAS STRAIGHT AT US. AT 7 MI; I STARTED A TURN TO THE S. CLOSURE RATE WAS SO RAPID; BY THE TIME I HAD TURNED 10 DEGS OFF COURSE TARGET WAS INSIDE 5 MI AND TCASII WAS DIRECTING A RAPID DSCNT. AT THAT INSTANT I HAD THE TARGET VISUALLY AND DETERMINED AN INCREASE IN TURN RATE TO THE S WOULD BE A BETTER COURSE OF ACTION. DURING EVASIVE ACTION I MADE THE RADIO CALL 'TCASII TA OUR ALT; ACR XXY HAS TFC IN SIGHT.' AFTER A PAUSE HE RESPONDED; 'TURN R HDG 180 DEGS;' THEN TOLD OTHER ACFT TO DSND TO FL370. PASSING THROUGH ABOUT A 150 DEG HDG I WATCHED THE OTHER ACFT PASS BEHIND US ABOUT A MI. IT APPEARED TO BE SOME TYPE OF CORPORATE JET; POSSIBLY AN MDT. HE HAD NOT BEGUN HIS DSCNT YET (VERIFIED BY TCASII). IT'S MY BELIEF THAT THE OTHER ACFT HAD NO IDEA OF OUR PRESENCE AND PROBABLY DID NOT HAVE AN OPERATING TCASII DUE TO THEIR INACTION; AND BECAUSE OF THE RADIO SILENCE PRIOR TO THE EVENT; THEN THE UNEASINESS IN THE CTLR'S VOICE AFTER I ALERTED HIM. I THINK IF WE HAD NOT HAD OUR TCASII; THE CHANCES OF SEEING THAT SMALL JET AND HAVING ENOUGH REACTION TIME IS DOUBTFUL. I THOUGHT THIS WAS ONE OF THOSE THINGS THAT JUST DIDN'T HAPPEN HAVING ALMOST 10000 HRS WITH NO CLOSE CALLS. I'M DEFINITELY CHANGING MY SKEPTICAL ATTITUDE TOWARDS THE VALUE OF TCASII AS MY CREW MEMBERS AND I AGREED THAT WE CAME WAY TOO CLOSE TO A MIDAIR KISS AT 39000 FT.