The mentally ill are less likely to be involved in a violent crime or shooting than the general population. Discrimination against them isn't based on evidence, and will not result in fewer shootings.
Also keep in mind that the weapon used in the Florida shooting was stolen from a locked safe.
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He didn't steal the gun, he owned it. The home owner made him keep it in a safe, but the shooter had a key to the safe
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You're correct, he bought it legally when he was a minor, and it was stored in a locked safe, which the homeowner presumed he had the only key to.