So we are allowing S to have more than 4 elements (although we dont need that in this case), so it is not just looking at a small number of factorizations of a 4 element set. This is because we want an FFS model, not just a factorization of the sample space.
If you factor in a different way, X will not be before Y, but if you do this it will not be the case that X is orthogonal to X XOR Y. The theorem in this example is saying that X being orthogonal to X XOR Y implies that X is before Y.
So we are allowing S to have more than 4 elements (although we dont need that in this case), so it is not just looking at a small number of factorizations of a 4 element set. This is because we want an FFS model, not just a factorization of the sample space.
If you factor in a different way, X will not be before Y, but if you do this it will not be the case that X is orthogonal to X XOR Y. The theorem in this example is saying that X being orthogonal to X XOR Y implies that X is before Y.