The second postulate is not needed. The first postulate says that all the "laws of physics" apply in an inertial frame, so that you cannot determine by any experiment. The speed of light (whatever it is!) in vacuo is given by Maxwell's equations, so the constant value of c is a consequence of the first postulate.
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"Who cares about half a second after the big bang;
what about the half a second before?"
- Fay Weldon
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Dr. George Spagna
Physics Department
Randolph-Macon College
P.O. Box 5005
Ashland, VA 23005-5505