Ok so I know how to prove Time Dilation using the Lorentz transformation...
My problem is that I don't see why one couldn't use the Inverse Lorentz Transform and come to the opposite result as follows:
Time period in rest frame S: T = t2 - t1
ILT: t = $(t1' + B/c x') where $ is gamma
So: t2-t1 = $(t2'-t1') so T' = T/$ which is the wrong result...
I.e. how does one know that one must use the LT rather than ILT?