 Just substituting into the Lorentz transformations we have, and then subtracting one from the other, we get this. So in other words, the interval delta xb transforms just like xb, and similarly for delta tb. So this is actually what linear transformations mean. The intervals transform just like the points. So you may ask, why are we assuming linear transformations? Well for starters, it seems like a good guess. I mean the Galilean transformations are linear, so maybe the relativistic ones are as well. Also the fact that distances transform just like points means these transformations aren't making any one point in space special. All that you have to care about are the relative distances between two points. Ultimately, however, this isn't so much something where I'm pausing on the system as rather a guess that we're making. We're seeing can a linear transformation work, so we'll try and fit it because we've got all these great reasons to think about why it should, and then hopefully it'll work out. If not, we'll try something else.