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where [tex]\sigma[/tex] represents a function from the reals to the manifold and f is a coordinate function from the manifold to the reals.

so i am starting with [tex]\sigma_{1} (0) = \sigma_{2}(0)[/tex]since i am picking two curves that go through the same point. i then define the functions [tex]F = \phi o \sigma_{1}[/tex] and [tex] G = \phi o \sigma_{2}[/tex] and since these are both functions from R to R my goal is to now show that F'(0) = G'(0). i also know that f(0) = g(0) since [tex]\sigma_1(0) = \sigma_2(0)[/tex] but now i am stuck and can't figure out the next step. can someone give me a few hints in the right direction? thanks.