Differential Geometry question

[MathStudent34]

I am looking for a rigorous proof for the following theorem (which I quote from Schaum Differential Geometry book):

 

"If K#0 at a point P on a surface, show that there is a neighborhood of P in which the points can be put into a 1-1 correspondence with the spherical image of the neighborhood (see Problem 9.57)".

 

Note: K is Gauss curvature.