Preliminary statement and formulae

In order to find the derivative (4.68) we fulfil a transformation of the domain fig, so that the domain is mapped onto fl. Let 9 G C D) be any function such that 0 = 1 in a neighbourhood of the point Xi = (/,0). To simplify the arguments the function 9 is assumed to be equal to zero in a neighbourhood of the point (0,0). Consider the transformation of the independent variables 

It is easy to find the derivative of Ag y) with respect to 5, namely, 

Assuming that y,5 are independent variables in (4.69) we have x = x y,5). Differentiation of (4.69) with respect to 5 yields 

In accordance with (4.69), let x = x y,5). Then w x) = ws y). The inclusion G Kg implies wg G Kq, and, conversely, wg G Kq implies G Kg. This means that the transformation (4.69) maps Kg onto Kq, and it is one-to-one. Now we shall prove an auxiliary statement which is used in the sequel. 

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