We have that
Since , we may view as
Consider the map
It is clear that is well-defined since if , then .
If , it is clear that . So is injective. It is also clear that is surjective.
Note that , where denotes the set of 2 by 2 complex matrices.
When is viewed as a function , it is clear that and are smooth maps since their component functions are of class . Since and are submanifolds, the restrictions to these submanifolds (i.e.\ and ) are also smooth.
Hence is a diffeomorphism.