Why does refraction of light involve a difference in speed?

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As I understand it, refraction of light works because of a difference in the speed of light between the two medums involved (Snell's Law?).

However according to the theory of relativity the speed of light is a constant.

These two statements seem incompatible and I would like to know if there is a simple explanation.
 
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The speed of light in vacuum is a constant.
 
In Relativity, it is the speed of light in a vacuum that is constant. Also, in a medium the reason the apparent speed of light is lower is that the light is being absorbed and re-emitted by the atoms of the medium. It is the delay between the absorbtion and emission that seems to "slow" the light down. While the light is traveling from atom to atom it still travels at the constant speed it does in a vacuum.
 
Many thanks to you both!

Very simple really!
 

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