My suggestion is to read thoroughly the proof. All the important steps are contained and there should be no need for any extra explanation.

The point is to decompose an arbitrary vector v into two parts, one in the image of P and another in the kernel of P. Thus, in the proof, we have the decomposition into two parts (the red-boxed one and the blue-boxed one) and we showed that the red one is indeed in the image, while the blue one is indeed in the kernel.

In the green box,  we show that the decomposition is unique in the sense that if we have two decompositions, they have to be the same.

Have you seen the proof I gave? Where do you find hard to understand?

We used the P^2 = P property several times in the proof.

ans