For a start, is a projection, with range . So if then , , and hence . Thus if is a projection then its range contains .

For the converse inclusion, if is a projection then

,

(since and ),

.

Multiply both sides of (3) on the left to get

.

Multiply both sides of (3) on the right to get

.

From (4), (5) and (3),

.

Therefore

.

It follows from (7) that the range of is contained in the range of and in the range of , and thus in .