  Weyl begins his proof with two natural assumptions. First, the  nature of the metric should be coordinate independent. If  \(ds\) is given by an expression  \(F_{p}(dx^{1}, \ldots ,dx^{n})\) with respect to a given system  of coordinates, then with respect to another system of coordinates,  \(ds\) is given by a function that is related to  \(F_{p}(dx^{1}, \ldots ,dx^{n})\) by a linear, homogeneous  transformation of its arguments \(dx^{i}\).  Second, it is reasonable to assume that the nature of the  metric is the same everywhere, in the sense that at every point of  the manifold, and with respect to every coordinate system for a  neighborhood of the point in question, \(ds\) is represented by an element of the equivalence class \([F]\) of functions generated by any one such function, say  \(F_{p}(dx^{1}, \ldots ,dx^{n})\), by all linear, homogeneous  transformations of its arguments \(dx^{i}\). 