ACR X TCASII TA RA. HAD NMAC; LTSS WITH ATX Y ON VISUAL APCH TO ANOTHER ARPT. SYS ERROR.

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ACR X ON A VISUAL APCH TO RWY 16R SEA. WE HAD A NEAR MISS WITH A PA31 PIPER NAVAJO; ATX Y. WE WERE AT APPROX 1000 FT AGL/1400 FT MSL. THE FO WAS FLYING AND COUPLED TO THE AUTOPLT ON THE RWY 16R ILS. ATC CALLED TFC WHICH WAS A COMMUTER ON SHORT FINAL TO RWY 16R. BOTH PLTS SAW THE TFC. WHEN PASSING OVER BOEING FIELD WE RECEIVED A TCASII TA FOR TFC WITH NO ALT RPTED. THIS IS COMMON AND WE DID NOT SEE THAT TFC. THE APCH CONTINUED AND WE RECEIVED ANOTHER TA FOR TFC 5 MI AT 11 O'CLOCK AND CLOSING. THIS SECOND TA WAS RAPIDLY FOLLOWED BY AN RA TO DSND. THE FO INCREASED DSCNT WHILE BOTH PLTS LOOKED FOR THE TFC. THE CAPT SPOTTED THE TFC VISUALLY. HE TOOK CTL OF THE ACFT AND BANKED SHARPLY L WHILE CONTINUING TO FOLLOW THE RA TO DSND. BOTH PLTS THEN SPOTTED THE PA31 TFC INITIATING A SHARP L BANK BTWN 500 FT AND 1000 FT IN FRONT OF US. AFTER THE CONFLICT HAD BEEN AVOIDED; WE ADVISED SEA TWR AND THE CAPT MANEUVERED BACK TO THE APCH PATH. WE MADE A NORMAL LNDG ON RWY 16R SEA. UPON ARR TO THE TERMINAL; WE DISCUSSED THE INCIDENT WITH SUPVR OF THE SEA TWR. THE WX WAS VFR. THE TCASII WARNING WAS FOLLOWED AND SAVED US FROM A POSSIBLE FATAL CRASH. WE WERE TOLD THE PA31 WAS ON A VISUAL TO BOEING FIELD.