Relativistic velocity addition law derived from a machine gun analogy and time dilation only

We consider a scenario that involves a machine gun, the bullets it fires and a moving target, considered from the rest frame of the machine gun and from the rest frame of the target respectively. Involving the special relativity via its two postulates and the time dilation formula we derive the relativistic velocity addition law showing that it leads to the Lorentz transformations for the space-time coordinates of the same event.