Attorney Steve Kelly, who's repping the Lee family, tells TMZ ... Hae Min's brood simply doesn't think what prosecutors submitted to cut Adnan loose was convincing, adding they don't consider it real evidence that actually points to another killer or killers.

Kelly says the Lee family was told for 20 years by the Baltimore State's Attorney's Office Adnan was their man -- responsible for Hae Min's death -- something a jury at the time agreed with when they convicted him of murder and sentenced him to life in prison.

Now we're told they feel they've had the rug pulled out from under them. Kelly tells us the Lee family would've absolutely consented to springing Adnan out of prison, IF they actually bought the arguments prosecutors made in their recent court motions.

FWIW ... prosecutors pointed to notes and/or letters from at least one potential suspect -- who was apparently known to cops from the jump and who apparently wrote they wanted to make HML "disappear" and "kill her." The problem, according to prosecutors, is the notes were not handed over to AS's defense team during the trial, which is a big no-no because any evidence that potentially exonerates AS should've been handed over.

Not just that, but prosecutors are also now saying a key witness -- Jay Wilds -- gave inconsistent accounts to cops.

Apparently, much of this stuff wasn't disclosed to Adnan's team ... and prosecutors are saying it all warranted throwing out his conviction. As we reported ... a judge agreed Monday and ordered Adnan be cut loose ... although he'll remain on house arrest for 30 days while prosecutors evaluate whether to retry him.

While some say this is significant evidence and more than enough to raise reasonable doubt  ... the Lee family believes there's not enough info to persuade them Adnan should go free.