Complete brainlet American here, been trying to learn linear algebra

1. given: an identity I(v), and a projection: P*P=P
2. since I(v) is an identity, and P is a projection, we can write: P*I(v)=P
3. by ker: P*(I(v)-P)=0
4. since P is a projection and by #3 we can write: P*(I(v)-P)=P*(I(v)-P), which is a projection
5. now since I(v) is an identity element and P*P=P, by #1 and #2, this implies: I(v)=P
6. by ker: I(v)-P=0
7. which we can write as: I(v)-P=I(v)-P
8. and by #4 this implies I(v)-P=P*(I(v)-P)
9. and since P*(I(v)-P) is a projection
10. then I(v)-P is a projection