If , are two projections in , then is the range projection of , and is therefore in By induction, the sup of any finite family of projections in is also in

Given an infinite family, the directed net of sups of finite subfamilies (ordered by inclusion) is increasing, and bounded above by the identity. Therefore it converges strongly to a limit which is a projection and is therefore the sup of the whole family.

The result for infs follows by taking orthogonal complements.