CLOSE NMAC BTWN A VFR CESSNA DEP NW FROM HWO AND A B757 ON APCH TO ILS RWY 9L AT FLL; FL.

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ON ARR TO FLL WE WERE ON A 240 DEG HDG AT 4000 FT ON DOWNWIND FOR RWY 9L FLL. WE WERE CLRED TO 2000 FT AND GIVEN HDG 180 DEGS. TOLD APCH CTL WE HAD FIELD IN SIGHT. WE WERE CLRED THE VISUAL; TOLD TO SQUARE TURN TO FINAL. ENTERED A DSCNT AND AS FINAL APCH COURSE WAS BEING INTERCEPTED; RECEIVED A TCASII 'TFC; TFC.' I LOOKED OUT THE WINDOW AND VISUALLY ACQUIRED THE CESSNA. COPLT DID NOT SEE ACFT BECAUSE CTR WINDSHIELD POST OBSTRUCTED VIEW. I INSTRUCTED COPLT TO LEVEL OUR ACFT. OUR ALT WAS 2100 FT; THE TFC SHOWED 1800 FT. APCH CALLED AND SAID 'POP UP TFC; 12 O'CLOCK POS' THEN THE TCASII SAID RA 'CLB; CLB; CLB.' THE COPLT FOLLOWED THE RA AND INITIATED A CLB. WE CLBED TO ABOUT 2300 FT AND WENT ACROSS THE RWY 9L FINAL TO RWY 9R. I VISUALLY CLRED THE CESSNA PAST US. WE TURNED TO REINTERCEPT THE RWY 9L FINAL AND CHANGED TO TWR. TWR QUERIED US AS TO WHY WE CLBED AND WE TOLD THEM IT WAS TO AVOID THE CESSNA. THEY SAID THEY WOULD 'FLAG' HIM AND FOLLOW UP. I FELT THAT WITHOUT TCASII THIS WOULD HAVE BEEN A NEAR MISS OR WORSE. AT THE ALT OF THE LIGHT ACFT 100-200 FT OFF OUR ALT; ABSENT THE CLB WARNING; HAD WE MISSED HIM; OUR WAKE TURB COULD HAVE CAUSED THEM TO LOSE CTL. THIS WAS DEFINITELY A SAVE DUE TO TCASII.