PROVIDING VISUAL SEPARATION FROM A PREVIOUSLY DEPING GULF; C560XL FLT CREW INITIATED TCASII RA MANEUVER DUE TO AN OVERTAKE; THEN RECEIVED ANOTHER TCASII ALERT DUE TO OVERHEAD XING TFC AS THEY SLOWLY DSNDED BACK TO ASSIGNED ALT WHEN DEPING KLAS.

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WHEN CLRED FOR TKOF AT LAS; THE TWR CTLR SAID 'RPT DEPARTING ACFT AHEAD IN SIGHT AND CLRED FOR TKOF. WE CLBED FASTER THAN THE ACFT AHEAD; AND WAS HIGHER; WHEN AT 6700 FT WE RECEIVED A TA FOLLOWED BY A RA. I LOOKED AT THE ACFT AHEAD; BECAUSE I INITIALLY THOUGHT IT WAS THE REASON. THEN I LOOKED AT THE FMS AND SAW A RED SQUARE WITH +17 SLIGHTLY TO OUR L. I LOOKED AT OUR ALT AND CAUTIONED THE CAPT TO WATCH OUR ALT; BECAUSE IT SHOWED 7100 FT AND CLBING SLOWLY. FOR WHAT I THINK WAS HIS CONCERN FOR PAX COMFORT; HE ALLOWED THE ALT TO GET TO 7200 FT BEFORE RETURNING TO OUR ASSIGNED ALT OF 7000 FT. THE ACFT PASSED OVER US AT NO LESS THAN 1500 FT ABOVE US. I DIDN'T TELL THE CTLR WE HAD A RA; EVEN THOUGH THE CAPT SAID TO ME 'WE HAD A RA' WHEN THE CTLR CLRED US TO CLB AND TURN R TO ANOTHER HDG; THEN ASKED US TO CONFIRM ON ALT; WHICH WAS 7400 FT. SHE STATED YOU CLBED 200 FT HIGH AND ASKED US TO COPY A PHONE NUMBER; THEN CLRED US TO ANOTHER FREQ. I MADE A MISTAKE IN NOT STATING IMMEDIATELY THAT WE HAD A RA. NEXT TIME; I WILL DO SO IMMEDIATELY. SUPPLEMENTAL INFO FROM ACN 577985: A GULFSTREAM CORPORATE ACFT WAS CLRED FOR TKOF 20 SECONDS PRIOR ON THE SAME RWY. AFTER TKOF AND CLBING THROUGH 6300 FT; A RA ADVISED INSTRUCTED US TO INCREASE VERT SPD. AFTER SEEING THE GULFSTREAM THAT CAUSED THE RA; WE IMMEDIATELY LEVELED OFF AT 7200 FT; AND THEN IMMEDIATELY DSNDED TO 7000 FT. THIS ALT DEV DID; HOWEVER; SET OFF ANOTHER TCASII RA ON AN ACFT OVERHEAD THAT WAS ON R DOWNWIND FOR RWY 9R AT LAS. WE FAILED TO NOTIFY ATC OF THE RA UNTIL LATER.