I'm not sure about your question nor your notation. But let me guess: suppose we are given vectorsxandysuch that the direct projection ofyonxgives a non-null vector. Now, I assume that, given a vectorylike this, it is possible to find a subspace V such thaty, when projected onto V, gives the null-vector. If so: we have rejected the above proposition, because no amount of further projecting that projection ofywill ever give somethingotherthan the null-vector.