Constancy of speed of light In a particular derivation of the Lorentz transformation, as illustrated in $\textbf{Relativity, Gravitation and Cosmology , Robert A Lambourne    }$ (p. 21) one of the key elements is the constancy of light.
So if $S$ and $S'$ are two inertial frames, with $S'$ moving with a constant velocity $V$ with respect to $S$, then in $S$ we have $c = \frac{x}{t}$ and in $S'$ we have $c = \frac{x'}{t'}$ even though $S'$ is moving with respect to $S$. 
My question is, what is this constancy of the speed of light based upon, other than it allowing for a simplification in the coordinate transformation? Was this experimentally confirmed?
The consequence of this leads to many strange postulates, such as the classic "moving clock runs slower". 
 A: There are strong theoretical arguments and immense experimental evidence for this constancy.
The form of the Lorentz transformation (i.e. that the group of transformations between inertial frames is the Lorentz group) with the speed $c$ as a free parameter follows from very basic symmetry assumptions about the universe. I discuss these further here.
In particular, in a universe with the above symmetries hold and where the value of $c$ is finite, it is readily deduced that anything moving at $c$ relative to some inertial observer is seen to be moving at $c$ relative to all inertial observers. Therefore, the observation of anything with this invariant speed behavior confirms experimentally the above theory.
The Michelson-Morley experiment famously succeeded in doing this, thus establishing experimentally that there is indeed an invariant speed $c$, and that light moves at this speed for all observers (or at least does so to an extremely good approximation). Modern versions of the Michelson-Morley experiment (using resonant interferometers rather like small versions of the LIGO ones) show that the change in the speed of light $\Delta c_L$ wrought by the addition of the Earth's motion is of the is of the order of $\Delta c_L/c_L<10^{-15}$. Incidentally, this also shows that light must be mediated by a massless particle (or again, one with exquisitely small mass).
Another property of this invariant speed is that, if a particle has nonzero mass, as we push harder and harder giving it higher and higher kinetic energy, its speed relative to us must asymptote to $c$. This is exactly what we observe in particle accelerators. In the LHC experiments, for example, we can treat all the collided particles - protons - as being massless, because their energy is thousands of times their rest energy (proton rest mass $\approx 1 {\rm GeV}$, LHC proton total energy $\approx 7{\rm TeV}$). It's not as accurate a way as the Michelson-Morely experiment to confirm the invariant speed's existence, but it is in some ways more intuitive and compelling.
