Differential Geometry question

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

"If [tex]K\ne0[/tex] 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.