NMAC OCCUR BTWN 2 ACR ACFT WHEN A B737 IS ALLOWED A VISUAL SEPARATION ON A PRECEDING B757 BUT LOSES SIGHT OF IT ON THE BOWIE ARR AT KAGLE INTXN; TX.

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ATC FACILITY: REGIONAL APCH CTL. LOCATION: 5 NM NW KAGLE ON BOWIE 7 ARR; ARR ON THE BOWIE 7N ARR. APCHING KAGLE FROM 5 NM TO THE NW WITH ATC CLRNC TO CROSS KAGLE AT 11000 FT AND 250 KTS. DSNDING THROUGH 11800 FT; RECEIVED FIRST SERIES OF TCASII EVENT RA MESSAGES -- 'MONITOR VERT SPD;' THEN 'MONITOR VERT SPD;' THEN 'DON'T CLB.' THEN 'DON'T CLB;' THEN 'DSND;' THEN 'DSND;' THEN 'DSND NOW.' THE FO FOLLOWED THE TCASII RA AND I NOTIFIED ATC OF THE RA EVENT. ATC ADVISED A FOLLOWING ACFT WAS MAINTAINING VISUAL SEPARATION; A B737. AS WE BEGAN TO LEVEL OFF AT 11000 FT; WE RECEIVED A SECOND SERIES OF A TCASII RA EVENT. ON THE 5 NM SCALE OF TCASII; A RED SQUARE +400 FT FOLLOWED TCASII RA INSTRUCTIONS; AND I ADVISED ATC OF TCASII RA AND OUR INTENTIONS. EXERCISING EMER CAPT'S AUTH; I ORDERED FO TO BREAK OUT R TURN OR S OF COURSE AS WE DSNDED TO GET LATERAL SEPARATION AND ACQUIRE VISUALLY THE OTHER ACFT. WHEN TCASII ADVISORY SAID CLR OF CONFLICT; I SPOTTED THE OTHER ACR B737 ACFT. WE REJOINED ARR AND CLBED TO RETURN TO ASSIGNED ALT; 11000 FT. I SPOTTED B737 ACFT DIRECTLY OVER THE TOP OF US AS WE BEGAN TO CLR THE ARR STREAM. AFTER SECOND TCASII RA EVENT SERIES AND LACK OF INTERVENTION BY ATC I COULD NOT TAKE THE CHANCE THE OTHER B737 ACFT DIDN'T REALLY HAVE OUR ACFT VISUALLY. CONDITIONS WERE: DAYTIME; CLR SKY; UNRESTR VISIBILITY. WHY DOESN'T TCASII GIVE LATERAL ESCAPE INSTRUCTIONS?