Can anyone clarify this example of Relativity of Simultaneity While doing this course: http://www.worldscienceu.com/ I couldn't understand this example http://www.youtube.com/watch?v=53oAxycVhhg. (full example here https://www.youtube.com/watch?v=Iiugtmt18W4)
As from what I understand, what makes a thing inside a moving environment behave as if they were at rest is the fact that the velocity of the environment will be "added" to whatever new velocity is generated in the environment (i.g. : throwing a ball, jumping etc..). However in that example since the speed of light is constant the people in the train and outside the train should see the exact same thing! Can anyone clarify why this reasoning is not correct ?
EDIT:
To avoid watching the video, I will try to explain the example:
In a moving wagon a light beam is fired in both directions (front & back)from the middle of the wagon. The people inside the wagon seen the light beam reach both ends of the wagon at the exact same time while an outside observer would see it rear end being hit first then the front end.
 A: 
Why both the observers should see the speed of light be exactly the same? 

To cut the long story short 'It is an experimental fact.'! The principle of relativity suggests that the laws of Physics must be the same in all the frames. Experiments suggest Maxwell's equations to be the correct laws of Physics. So a theorist can argue that everything that is suggested by Maxwell's laws should be true in all the frames. One of such things is that the speed of light is $\dfrac{1}{\sqrt{\mu_0\epsilon_0}}$. This must be true in all the frames. But, of course, nature can always surprise us so the ultimate test of the validity of this claim is the experiment. And it has been verified (to great accuracies) that the speed of light doesn't depend on the frame. 
Taking this invariance of the speed of light to be an axiom, as you have mentioned, one can conclude that the simultaneity is relative. 
Another point to note is that if the true law of addition of velocities is employed then one can conclude from the example of throwing regular objects like balls that the simultaneity is relative. There is nothing so special about light fundamentally (In this context). It is just that even before knowing the complete velocity transformation law, Einstein had known how the speed of light transforms between frames : It remains invariant. And thus, in the development of the theory of relativity, light related gedankens were used to reach the conclusion that simultaneity is relative.  
