We now prove that the rules given in section , for constructing a chart, are correct.
The forbidden region is clearly correct since is a function; we simply decree that as our use of the chart does not depend on how such points are treated. For (x,y) in the zero region, and ; so for any point in the zero region , which implies .
For the remaining regions, consider the polynomial
Consider the point (x,y) which is away from :
Earlier we proved q(x) interpolated for any m. We have now shown q(x) interpolates . The leading coefficient of q(x) is m. The sign of m relates to : the sign of m is positive if the region (x,y) resides in is labelled with ; the sign of m is negative if the region (x,y) resides in is labelled with .
|Jeff Tupper||March 1996|