Contract/Cap Question for BAB

With everyone speculating about what The Org will be able to accomplish this summer with regard to free agents, Tyrus' extension, etc., I'd be interested to know the answer to this question:

Let's say (hypothetically) that the Bulls are able to create enough cap space to sign a max free agent without moving Tyrus or renouncing his rights, and the first thing they do this summer is sign Wade.  Obviously this assumes that they have been able to create enough cap space around Tyrus' cap hold to sign Wade (probably not possible, but let's assume it is for now).  Once the ink is dry on Wade's contract, can they then go ahead and sign Tyrus to a contract greater than his cap hold, thereby going above the cap due to the fact that the Bulls still own Tyrus' rights (and you're allowed to go above the cap to resign your own players)?

If this is legit, then this only works in that order: sign Wade first (so that total payroll is under the cap), then sign Tyrus (and legally go above the cap).

In reality, this is probably a moot point, as Tyrus' cap hold is likely to be greater than he is worth anyway, and the Bulls are unlikely to make the moves necessary to free up max FA cap space around his cap hold.  However, if Tyrus has an underwhelming or injury plagued year - or the market for his services is quiet - maybe the bulls could sign him to a highly discounted deal (well below his cap hold), and still have room for Wade/Bosh/Johnson.  In this scenario though, I think we're more likely to see Taj Gibson as starting PF rather than resigning Tyrus the rabble rousing trouble maker.