that may be true, but remember my motivation for providing a proof. i was not sure if it was direct consequence, so i offered a proof to let you guys see what i was thinking why it is a direct consequence, so you guys could tell me if i was thinking along the right path. and besides, mathematicians are very precise, even corollaries and direct consequences have to be proven at some point before they are used, the mathematician must show WHY they are corollaries and consequences