B737 DEP FROM HOU EXPERIENCED TCASII RA IMMEDIATELY AFTER TKOF.

Narrative

WE WERE OPERATING AT HOU. WE WERE GIVEN POS AND HOLD ON RWY 4 TO EXPECT AN IMMEDIATE TKOF DUE TO ACFT ON A 3 MI FINAL. TWR TOLD SOME ACFT ABOUT US DEPARTING SHORTLY AND THIS ACFT STATED THAT HE SAW US IN POS ON THE RWY. I PRESUME THAT THIS WAS THE ACFT THAT WE SOON WOULD HAVE A CONFLICT; BUT AM NOT SURE. A DIFFERENT CTLR THEN GAVE US TKOF CLRNC FROM RWY 12 WITH A HDG OF 140 DEGS. THE FO RADIOED TO CONFIRM THAT THIS WAS REALLY FOR US. AN APPARENT SUPVR BROKE IN AND CLRED US TO TAKE OFF FROM THE CORRECT RWY (RWY 4) AND AN INITIAL HDG OF 170 DEGS. OUR INITIAL CLRNC ALT WAS 5000 FT. NEITHER OF THE CTLRS MENTIONED ANY CONFLICTING TFC. AT 400 FT AGL; WE STARTED OUR TURN. AT ABOUT 800 FT AGL; SIMULTANEOUSLY; WE RECEIVED A CALL FROM TWR ABOUT CESSNA TFC; WE ACQUIRED THE ACFT VISUALLY AT 2 O'CLOCK AND 300 FT HIGH AND RECEIVED A TCASII RA TO MAINTAIN VERT SPD. WE ROLLED OUT OF OUR TURN AND SAW THE CESSNA IN A STEEP 60 DEG BANK TRYING TO GET OUT OF OUR WAY. AFTER CLRING; WE RPTED OUR RA TO TWR AND WENT TO DEP CTL. AFTER LNDG; I CALLED HOU TWR AND TALKED TO A SUPVR. HIS INITIAL REACTION WAS THAT THE CESSNA HAD RPTED US IN SIGHT; AND; THEREFORE; IT WAS HIS RESPONSIBILITY TO MAINTAIN SEPARATION. I DO NOT THINK THAT IT WAS REASONABLE FOR THE CESSNA TO BE ABLE TO MAINTAIN SEPARATION FROM A JET TRANSPORT BEING VECTORED DIRECTLY AT HIM AND THROUGH HIS ALT; TO BE ABLE TO DO SO. IF WE HAD NOT ROLLED OUT OF OUR ASSIGNED TURN; WE WOULD HAVE BEEN VERY CLOSE.